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OU 162959 



OSMANIA UNIVERSITY LIBRARY 

Call No. ^'x >0 j Accession No. 3> 

Author 

Title 

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ELEMENTS OF WAVE MECHANICS 



ELEMENTS OF 

WAVE MECHANICS 



BY 

N. F. MOTT 

M.A., D.Sc., F.R.S. 

Cavendish Professor of Experimental Physics 
University of Cambridge 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 
1958 



P IT B L I S II K D BY 
T H TC SYNDICS OF T TI K CAMBRIDGE U N I V K R S I T Y 1* R E S S 

Bentley House, 200 Eastern Road, London, N.W. 1 
American Branch: 32 Kawt 57th Street. New York 22. N.Y. 



First Printed 1952 
Renrinfed 1958 



First printed in Great Britain by John Wright & Sons Ltd. , Bristol 
Reprinted by off set -lit ho by Bradford and Dickens, Northampton 



PREFACE 

This book is written to replace the author's An Outline of 
Wave Mechanics, published by the Cambridge University Press 
in 1930 and now out of print. It is intended for students in the 
final year of an honours course of experimental physics, and 
also as an introduction to more advanced text-books for those 
who intend to specialise in the subject. With such readers in 
view, and in order to keep the book reasonably short, 1 have 
not hesitated to omit some mathematical developments that 
can easily be found in other text-books. The book attempts to 
build up the elementary theory from experimental facts, and 
to show how simple problems can be solved. A certain number 
of examples are included. 

N. F. MOTT 

BRISTOL 

January, 1951 



CONTENTS 

CHAPTER PACK 

I. THE DIFFERENTIAL EQUATIONS OF WAVE 
MECHANICS 

1. Differential equations of the seeond order 1 

2. Wave equations 6 
8. Dispersion, group velocity and wave groups 1 2 

4. Fourier's theorem and characteristic functions 15 

5. Legendre polynomials 19 

II. THE WAVE EQUATION OF SCHRODINGER 

1. The wave function 21 

2. Sehrodinger's equation 25 

3. Bending of a beam of electrons 27 

4. Solutions of Sehrddinger's equation and the conser- 

vation of the number of particles 27 

5. The current vector 31 

0. The tunnel effect 32 

7. Scattering of beams of particles by atoms 36 

8. Electrons in a magnetic field 42 

III. WAVE GROUPS AND THE UNCERTAINTY 
PRINCIPLE 

1 . The frequency of electron waves 43 

2. The wave equation for non-stationary phenomena 45 

3. The description in wave mechanics of a single par- 

ticle 47 

4. The uncertainty principle 50 

IV. STATIONARY STATES 

1. The old quantum theory 52 

2. Treatment of stationary states in wave mechanics 55 

3. The simple harmonic oscillator 58 



Vlll CONTENTS 

CIIAPTKH PAGE 

4. Quantisation in three dimensions 61 

5. Quantisation with spherical symmetry and the 

hydrogen atom 62 

6. Interpretation of the wave functions 67 

7. Variational method of obtaining approximate 

wave functions H9 

8. Orthogonal property of the wave functions 70 

9. Perturbation theory 78 

10. The polarisability of an atom 74 

11. The Stark effect 78 

12. Effect of a magnetic field 79 

13. The electronic spin 81 

V. THE MANY-BODY PROBLEM 

1. The wave equation for two particles 84 

2. A pair of particles in one another's field 86 
2*1. The hydrogen atom, p. 87. 2-2. The diatomic mole- 
cule ', p. 88. 2-3. The France-Condon principle, p. 91 

3. Systems containing two or more particles of the 

same type 92 

4. Applications of the symmetry properties of the 

wave functions 06 

4-1. The exclusion principle, p. 90. 4-2. Particles 
ivithout spin obeying Einstein- Hose statistics, p. 96. 
4-3. Particles with spin obeying Fermi-Dirac statistics 9 
p. 97. 4-4. The rotational states of H 2 , p. 98. 4-5. The 
helium atom, p. 99. 4-5. The structure of atoms ivith 
more than two electrons, p. 102 
6. Interatomic forces and the formation of molecules 106 

6. Covalent forces 107 
6-1. The molecular ion H 2 +, jt. 108. ($-2. The method of 
molecular orbitals for flic hydrogen molecule H 2 , p. 112. 

6-3. The method of London-It eitler as applied to H 2 , p. 112. 
6-4. Some general features of chemical binding, p. 113 

7. The theory of solids 1 14 
7-1. (Concept of a conduction band, p. 114. 7-2. Semi- 
conductors, p. 116. 7-3. Metals, p. 118 



CONTENTS IX 

CHAPTER PAGE 

VI. TRANSITION PROBABILITIES 

1. General principles 121 

2. Excitation of an atom by a passing particle 123 

3. Transitions to unquantised states 128 

4. Transitions due to a force on an electron which is 

periodic in the time 129 

5. Emission and absorption of radiation 132 

6. Calculation of the B coefficient 134 

7. Selection rules 137 

8. Photo-electric effect 138 

9. Transitions due to a perturbation which does not 

vary with the time 139 
9-1. Anger effect, p. 139. 9-2. Scattering of electrons by a 
centre of force, p. 142 

VII. RBLATIVISTIC AND NUCLEAR DEVELOPMENTS 

1. Dirae's relativistic wave equation 143 

2. The positron or positive electron 144 

3. Applications to nuclear physics 146 
3-1. The theory of ot-decay of radioactive elements, p. 147. 

3-2. The theory of fi-decay, p. 150. 

BIBLIOGRAPHY 158 

INDEX 155 



CHAPTER I 

THE DIFFERENTIAL EQUATIONS OF 
WAVE MECHANICS 

1. DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 

Many problems in wave mechanics can be reduced to the 
solution of a differential equation of the type 

2> = > (1) 

and a thorough understanding of this equation is essential to 
the student of the subject; f(x) is a known function of *r, and 
by means of the equation it is possible to plot y against x when 
the values of y and of dyfdx are given for one arbitrary value 
of x. An equivalent statement is that two independent solu- 
tions, y^ and y 2 , exist and that Ay l + By 2 is the general solu- 
tion. Methodsf exist of obtaining these solutions graphically, 
which can be utilised when f(x) is given as a plotted curve or 
tabulated. 

The simplest case of equation ( 1 ) is that in which f(x) is a 
constant; if f(x) is a positive constant, we may write 

/(*) = *, 

and two independent solutions then exist, cos kx and sinkx. 
The general solution is 

y = A cos kx + B sin kx 

or y = a cos (kx 4- e), 

where A, B, a and are arbitrary constants. The solution is 
then oscillating (Fig. la). 

fCf., for example, D. R. Hartree, Proc. Manchr. lit. phil. Soc. LXXVII, 91, 
1933; W. F. Manning and ,7. Millman, Phys. Rev. Lin, 673, 1038; M. V. 
Wilkes, Proc. Camb. phil. Soc. xxxvi, 204, 194O; L. Fox and E. T. Goodwin, 
Proc. Camb. phil. Soc. XLV, 373, 1949. 



2 EQUATIONS OF WAVE MECHANICS 

If f(x) is constant and negative, we set 
f(x) = -/ 

and the solutions are then e~? x and e? x , with the general 
solution 

y = Ae? x + Be~? x . 

These solutions also are illustrated in Fig. 16. 



f<x) 







(B) 



Fig. 1 . Solutions of the differential equation y" +/(#) y = 0. (a) for 
f(x) = A; 2 , (6) for f(x) = y 2 , (c) for an arbitrary form of f(x) which 
changes sign, the solution for negative values of x taking either of 
the forms shown in (6). 

In the general case where f(x) is not a constant, it is easy to 
show that, if f(x) is positive, y is an oscillating function, while 
iff(x) is negative, y is of exponential form. For if f(x) is positive, 
y and dPy/dx 2 have the opposite sign. Therefore, if we consider 
any point A on the curve for which y is positive, the curve 
bends in the direction shown in Fig. 2a, getting steeper and 
steeper until it crosses the axis when it will begin to bend in 
the opposite direction. If, on the other hand, f(x) is negative, 
y and d 2 yjdx 2 ate of the same sign, and the slope at a point 
such as A will increase, giving an exponentially increasing 
curve as shown in Fig. 26. 



EQUATIONS OK SECOND ORDER 3 

The general form of the solution y for a function f(x) which 
changes sign is as shown in Pig. Ic. When f(x) becomes negative 
y goes over to the exponential form. It should be emphasised 
that there will always be one solution which decreases expon 
entially, but that the general solution will increase. If one 
stipulates that the solution should be that which decreases 
exponentially, this determines the phase of the oscillations in 
the range of x for which oscillations occur. 




X 

\ 




(*>) 



Fig. 2. 



A useful method exists of determining approximate solutions 
of the differential equation (1 ), for the case where f(x) does not 
vary too rapidly with x. This is known in the literature as the 
Wentzel-Kramers-Brillouinf (W.K.B.) method, though appar- 
ently first given by Jeffreys.J In the discussion given here we 
shall confine ourselves to the case where f(x) is positive. 

In order to obtain this approximate solution we set 



y = 



(2) 



Here a and /3 are both functions of #; a represents the amplitude 
of the oscillations, ]8 the phase. On substituting into (1) we 
obtain 



f G. Wentzel, Z. Phys, XXXVIH, 518-29, 1926; H. A. Kramers, Z. Phys. 
xxxix, 828-40, 1926; L. Brillouin, C.R. Acad. Sci., Paris, CLXXXIII, 24-6, 
1926. 

* H. Jeffreys, Proc. L&nd. math. Soc. (2), xxm, 428-36, 1925; Phil. Mag. 
xxxin, 451-6, 1942. 



4 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

where a dash denotes differentiation with respect to x. We are 
at liberty to impose one arbitrary condition on the two func- 
tions a, j3; we set therefore 



so that j8 - \f*dx, (3) 

J Xo 

where X G is an arbitrary constant. This gives us correctly the 
phase of the solution and in fact tells us nothing that we did 
not know already; if f(x) is varying slowly, y goes through a 
complete oscillation in an interval Ax of x such that /*A# = 2?r. 
The purpose of the approximate method explained here, 
however, is to estimate a, the amplitude of the oscillations. 
Since f(x) is varying slowly, so is a, and we thus neglect a" in 
comparison with od . We thus obtain 



whence on integrating 

Ina + lnj3' = const. 

This gives a = const. (jS')~*, 

or, substituting for /?' from (3), 

a = const./"*. 
The approximate solution (2) is thus 

f 
y - const./"* exp \i 



f J ' } 
f*dx\. 

Jar, ' 



It follows that the amplitude of the oscillations increases as / 
becomes smaller and the wavelength increases. This is shown 
in Fig. Ic. 

Exercise 

Show that if / = #~ 4 , the above solution is exact. 



DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 5 

111 certain cases it is possible to obtain a solution of (1), or 
of the more general equation 



in the form of a power series. As an example of the method 
we give the solution of Bessel's equation 

Idy 



We set for y y x^(a Q -f a^x -f 2 a;2 +) (&) 

When this series is inserted in the equation, the coefficient 
of every power of x must vanish. The lowest power of x which 
occurs is that of p 2; the coefficient of x p ~ 2 must vanish. This 
gives the quadratic equation known as the indicial equation 

p*-n 2 ^ 
or p = n. 

There are thus two possible values of /o; these give the two 
independent solutions of the equation. 
The coefficient of x?+* is 



Since this must vanish we obtain a relation between a s f 2 and a 8 . 
Thus from a we obtain a 2 ; from a 2 we obtain a 4 ; and so on. 
The student will easily see that the odd coefficients a l9 a 3 , etc., 
vanish. 

It is of interest to discuss the form of the solutions of (4) for 
large n by means of the W.K.B. method. The equation (4) may 
be reduced to the standard form (1 ) by means of the substitution 

y = ar*z, 
giving for, 



6 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

It will be seen that for x > ^l(n 2 - ) the solutions are oscillating, 
and for large x behave like cos a; and sin x or e lx . For 
x<*J(n 2 -- 1), on the other hand, they are exponential. The 
solution in series for which the first term in (5) is x 11 clearly 
corresponds to the solution which decreases as x decreases, 

while that which begins with 

x -n corresponds to the in- 
_ creasing solution. The two 

solutions are illustrated in 

Fig. 3. 




5 10 15 20 ~ 

Exercise 

Discuss the form of the 
solution for small positive 
values of x of 



i-o 



both by means of the indicia I 
equation, and by the W.K.B. 
method. Show that the num- 
ber of oscillations between 

05 \ x = and any finite value 

of x is infinite. Why is this 
' ^ ^ ^ ^ ^ ^ --"- ^ - case for tlie equation 



Fig. 3. The two solutions of (4) for n = 2. 2" + I 1 -f - 1 2 = ? 

2. WAVE EQUATIONS 

In most forms of wave motion in one dimension, the dis- 
placement T of the vibrating medium at a point x in space 
satisfies an equation of the form 



- . (O) 

a* w l ' 

v (the wave velocity) is a constant which depends on the 
medium. Examples are: 



WAVE EQUATIONS 7 

(a) The vibrations of a string of tension T and mass p per 
unit length. *F is here the lateral displacement, and x the 
distance measured along the string. The proof of this relation, 
together with the formula for v, 

v 2 = T/p, 

can be found in any text-book on vibrations, e.g. Coulson's 
Waves, Chapter II. 

(6) Sound waves. Here *F is the condensation, or Ap/p, where 
p is the density. 

(c) Electromagnetic waves. In a plane polarised wave, the 
state of the medium is described by the electric vector E and 
the magnetic vector H, which are perpendicular to each other 
and to the direction of propagation. When the latter is along 
the ar-axis they satisfy the equations 



8E BH 

f> _ --. __ __ ___ -.. _ 

8x Bt ' K fa dt ' v ' 

where c is the velocity of light in a vacuum and K the dielectric 
constant. Eliminating H between these equations we obtain 

( ' 

The general solution of equation (6), as may easily be veri- 
fied, is 



where F and G are any arbitrary functions whatever. This 
solution means that some quite arbitrary form of displacement, 
F(x), is moving with velocity v to the right without change of 
form, and another function G(x) moves to the left. This is 
illustrated in Fig. 4. 
Particular solutions are: 



(i) T 

which represents a simple harmonic wave of wavelength 
A = 27T/A;; 



8 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

(ii ) T = A {sin k(x -vt) + sin k(x + vt)} 

= 2 A sin kx cos fcttf , 

which represents a standing wave. 

The generalisation to three dimensions of equation (0) is 



where V 2 is defined as 

V 2 = 



Fig. 4. General solution of equation (6). 

The solution representing a plane wave moving in the direction 
defined by the direction-cosines (Z, m, n) is 

* = A sin {&(fo -f my + wz - trt)}. 

It is easily verified that this is a solution of (9), making use of 
the relation / 2 -f m 2 + n 2 1 . 

The wave velocity v may vary from point to point in space. 
Thus in the case of the vibrating string the density p may vary 
along the length of the string; in the case of electromagnetic 
radiation, the light may pass from a medium of one refractive 
index to another. Two cases of interest present themselves, 
(i) that of a sharp boundary and (ii) that of a gradual change. 

(i) There will be a sharp change in v, for example at the 
boundary between glass and air, or at a point where two 
strings of different density are joined together. Any wave inci- 
dent normally on such a boundary will be partially reflected. 



WAVE EQUATIONS 9 

We shall now calculate the amplitude of the reflected wave at 
a boundary where the wave number changes suddenly from k 
to k f . It is convenient to use the complex exponential form 
for the function representing the waves, it being understood 
that the wave amplitude is actually represented by the real (or 
imaginary) part of the function written down.f We then write 
for the incident wave! 

T 



for the reflected wave Ae i( ~ kx ~ Mt \ 
and for the transmitted wave 



The problem is to calculate A and B. a)/2ir is here the fre- 
quency, which, of course, cannot change when the wave goes 
from one medium into another. 

It is convenient to choose the plane x for the boundary 
at which v and hence k( = wfv) changes. We then have for the 
wave amplitude 

\p = ( e ikx _|_ A e~ ikx ) e-*** x < 0, 

= Be lk ' x e~- itot x>0. 

The values of A and B are determined by the boundary condi- 
tions applicable at x = 0. For the case of the string, it is 
obvious that these are that MP* and cW/dx must be continuous; 
in other words, there is no kink in the string. In the case 
where T is the electric vector E of a plane polarised wave 
falling normally on a reflecting surface, it may be shown that 
the same conditions are satisfied. The condition that T is con- 

tinuous then gives 
* 



and that S^jdx is continuous, 



f As will be shown in Chap. II, in wave mechanics we actually represent 
the amplitude by a complex function. 

J Alternatively we could use the complex conjugate; in this chapter w* 
shall use the convention that the time factor is c-**. 

Cf., for example, .1. A. Stratton, Electromagnetic Theory, New York, 
1041, p. 35. 



10 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

On solving for A and B we find 

A = (k- k')l(k + fc'), B - 2fc/(i 4- *'). 

If we define by R the proportion of the incident energy re- 
flected, then R can be equated to A 2 , so that 



Since &'/ = M where /i is by definition the refractive index, 
this may be written 

#=(1- M )2/(1 + /X) 2 . 

For glass, for example (/x~ 1-5), this gives /2~0-04. 

Exercise 

Verify, for waves in strings and for electromagnetic waves, 
that the flow of energy in the reflected and transmitted waves 
is equal to that in the incident wave. 

A particularly interesting case arises when k', the wave 
number in the medium on which the wave is falling, is imagin- 
ary. This can arise, for instance, for electromagnetic waves in 
a medium containing free electrons; if N is the number of such 
electrons per unit volume, the refractive index \ is given byf 



where e is the electronic charge; for low enough values of the 
frequency v the right-hand side becomes negative. For such 

cases we may write . , 

J *' = ty, 

where y is real and positive. For the transmitted wave (x > 0) 
we have then two alternative forms 



Obviously the latter is not admissible, since it increases 
indefinitely as x increases. We therefore take for x > 

T = J3e-i*e-H 

f This equation is discussed further in ('hup. IV, 10. 



WAVE EQUATIONS 11 

The boundary conditions now give 



ik(l-A) = -y, 

, A fe-iy 

whence A = 7 . . 

k + iy 

The modulus of A is unity; in other words, A is of the form 
e iot \ thus the amplitude of the reflected wave is now unity. 
Therefore, when a wave is incident on a medium of imaginary 
refractive index, it is totally reflected. 

Exercise 

Investigate the reflection of waves from a slab of material 
in which the refractive index is imaginary. Take for the 
boundaries of the slab x = and x = a, and set 

-ikx) e -i<,>t X< Q 

'*)e- iMi 0<a;<a, 

a<x. 
Show that if e~f a <g. 1, then approximately 



and thus that the intensity penetrating the slab decreases as 
e -2ya a8 a j s increased. 

(ii) If there is no sharp change in k, but a gradual change from 
point to point, then no partial reflection occurs; but a beam of 
waves will be bent as it traverses the medium. A formula for 
the radius of curvature of a beam of waves in traversing a 
medium will be important. Fig. 5 shows such a beam. ABC, 
A'B'C are wave fronts one wavelength apart. We require to 
know the radius of curvature p of the beam, equal to CO or CG'. 



12 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 



Let us denote by A the wavelength in the centre of the beam, 
shown in Fig. 5 by GG'. Then, if 2t is the thickness AB of the 

beam, we may write 



AA' = X+~t, 
vn 

where S/Sn means differentiation normal 
to the beam. Then applying the rules 
of similar triangles to the triangles CGG' ', 
CAA', we see that 

I 1 8X 




Fig. 5. 



This formula will be used further in 
Chapter II, 3. 



3. DISPERSION, GROUP VELOCITY AND WAVE GROUPS 

In wave motion of the type described by equation (6) there 
is no dispersion; that is to say the wave velocity v is inde- 
pendent of frequency. If, on the other hand, v is a function of 
frequency, dispersion will occur. 

We require first to prove the formula for the group velocity 
V G with which a group of waves propagates itself. This formula 

is j 

dto 



where o> = "2-nv and k = 2ir/A. The formula is often written 

dv 



It will be noticed that since w = vk, for non-dispersive systems 
the wave and group velocities are the same. 

The simplest way of obtaining formula (11) is to superimpose 
two simple harmonic waves with wave numbers A*, k' differing by 
a small quantity. We have then for the resultant amplitude 

sin (kx cot) -f sin (k'x a//.), 



DISPERSION AND GROUP VELOCITY 13 

which is equal to 

rt . (k + k f cu + o/ \ ik-k' co-o/ \ 
2 sin ^ ~x - - T - tj cos ^- x - * ) - 

This function represents a series of wave groups, as shown in 
Y"ig. fta, each of length Az = 27r/(&- &'), moving from left to 
right with velocity (<j*-a>')l(k-k'). Since fc-fc' and co-a/ are 
small, this may be written dajjdk. 






:' -A/V- - 

Fig. 0. 

Wave groups of length Ao: have thus been obtained by super- 
imposing two simple harmonic waves for which the values of k 
are separated by A&, where 

AzAA; 277. (12) 

It is of interest to prove a similar theorem for a single wave 
group. Suppose that we add together simple harmonic waves 
with wave-numbers lying in the range At by setting for the 
wave amplitude 

*F= f" A(k)e ikx dk 9 (13) 

J-OO 

where A(k) vanishes, or tends rapidly to zero, outside a range 
AJk about the value k . It is convenient to take for A(k) the 
Gaussian function, a form which permits the integrations to be 
carried out. We set 



14 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

The integral then becomes 



where a = 



The integral is easily evaluated; it may be written 



Putting k \bla = , and remembering that 



r 

J-c. 



we find for (14) 

or, in our case, \rr^ Afcexp [ (| &k) 2 x 2 -f ik Q x] . 
This represents a wave group of the form (Fig. 6b) 

const, exp { (#/| A.r) 2 -f ik x}, 
where, equating JAfc to 1/iA.r, we have 

; 8. (15) 



The somewhat different interpretations given to A.r, A& account 
for the difference between (12) and (15). 

It is of interest to generalise the integral (13) to investigate 
motion of the wave group, and thus to obtain the formula for 
the group velocity for a single group. At a subsequent time t 
the amplitude T will be represented by 



/*00 

J- 



CHARACTERISTIC FUNCTIONS 15 

If we expand 



then the integral may be evaluated as before. The reader will 
easily verifyf that the wave group moves with velocity dwjdk, 
and that the width A# increases, and becomes for large t 

AAr. 



4. FOURIER'S THEOREM AND CHARACTERISTIC FUNCTIONS 

It will be convenient to introduce the reader to the problem 
of characteristic functions by writing down again the differ- 
ential equation of a vibrating string (cf. equation (6)) 



_ 

a* ~ ar 2 ' 

where T is the displacement at a point distant x from one end 
of the string. We consider in this section the possible vibra- 
tions when the string is rigidly held at the two ends, for 
example at x = and x = a. 

We define as a 'normal mode' a mode of vibration of the 
string in which each point executes simple harmonic motion 
with frequency o>/27r, say. Thus for a normal mode 

Y(a; , t) = ^(ar) {A cos wt + B sin <*>*}, (17) 

where ifj(x) satisfies the equation, 

&- 

If the string is uniform, so that v is not a function of x, the 
solutions which vanish at x = are 

i/r = sm(a)X/v). (19) 

t Cf. N. F. Mott and H. S. W. Massey, Theory of Atomic Collisions, 1949, 
p. 17. 



16 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

The displacement of the string must also vanish at x a . ; this 
will be the case if 

atD/v = Tin, (20) 

where n is an integer. These values of w, then, determine the 
frequencies with which the string can vibrate. Substituting (19) 
and (20) into (17), we see that the most general type of vibra- 
tion of which the string is capable is that which results when 
all the normal modes are superimposed, namely, 



X sin (irnx/a) {A n cos a) n t + B n sin a>,, }, (21) 

n 

where A n1 B n are arbitrary constants and 



We shall now show how to determine the subsequent motion 
if the string is given any arbitrary initial displacement and 
released from rest. At time t = 0, then, let the displacement be 

y = TO, 

where F is some function that vanishes at x and x = a. 
Since (21) is the general solution of the differential equation 
(16), that is to say, it represents the most general motion 
possible, it must be possible to represent the subsequent motion 
by a series of this type. Since the string is released from rest 
at time t = 0, the coefficients B n must all vanish; the subse- 
quent motion is thus given by 

y = 2 A n cos w n t sin 1 1 . 

n \ a ' 

It follows that, putting t = 0, it must be possible to represent 
the function F(x) by an expansion of the type 

(22) 



An expansion of this type is known as a Fourier series. 



CHARACTERISTIC FUNCTIONS 17 

The coefficients A tl may be determined by making use of the 
orthogonal relation 



sin I 1 sin I ~ I dx 0, m 4= n, 

\ a / \ a / 

which is easily verified. It may also be seen that 

f . /7rnx\ 

sm 2 I \dx = } 2 a. 

Jo \ a / 

If then we multiply both sides of (22) by sin (irmxla) and inte- 
grate from to a, all terms vanish except that in w; we have, 
therefore, 



i f a x 
\aA m = jf'V) 



sm 



This equation determines the coefficients and hence the subse- 
quent motion. 

Exercises 

(1) Determine the coefficients A n for a string plucked at the 
centre, so that the initial displacement is given by 



Find the energy of each normal mode, and verify that the sum 
of the energies of all the normal modes is equal to the work 
done in displacing the string in the first place. 

(2) A string of mass per unit length />, tension T and length 
2a, is rigidly fixed at the two ends x = a. It is set in vibration 
by a sound wave which exerts on it a force p cos wt per unit 
length. Write down the equation of motion of the string, and 
verify that the solution corresponding to forced vibrations is 



y _. 

* [ 
where y is the displacement. 



p 

* 

pa)* cos 



IS DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

Write down also the complete solution of the equations, and, 
without working out any integrals, find the solution appropriate 
to the case where the string is at rest and undisplaced at 
time t = 0. 



The Fourier expansion is a particular case of a more general 
type of expansion, which may be illustrated by considering the 
normal modes of a vibrating string of which the mass per unit 




Fig. 7. 

length (/>) is not constant. It will be remembered that v 2 = Tjp\ 
v z will thus be a function of #, and we may write l/v 2 = /(#), 
so that (18) becomes 



W-<>. (23) 

Such an equation, together with the boundary conditions that 
if/ must vanish at x = and x = a, defines a series of values of 
o>. For suppose we choose a small value of co and obtain the 
integral of equation (23) that vanishes at x = 0; the solution, 
oscillating on account of the considerations of 1, will cross 
the rr-axis again at some value of x greater than a. As to is 
gradually increased, a function \fj will be obtained which does 
vanish at x = a, as shown in Fig. 7. We call this the first 
characteristic function, and denote it by ^(x), and the corre- 
sponding value of to by toj. Similarly $ 2 (x) denotes the solution 
with one zero between x = and x = a, and so on. 



LEGENDRE POLYNOMIALS 19 

Exercises 

(I) Prove the orthogonal relation 



f a 

Jo 



for two characteristic solutions of (23), and hence show how to 
expand any arbitrary function in a series of characteristic 
functions. 

(2) Use the W.K.B. method to determine the values of cu n 
corresponding to solutions of (23) for large values of r?. 

5. LEGENDRE POLYNOMIALS 

The Legendre polynomials P;(cos#) are defined as follows: 
If |#|<1 we may expand the quantity (1 2#cos0-f # 2 )~* in 
ascending powers of x. The polynomials are defined as the 
coefficients in the expansion. Thus 

1 = 1 + xP^cos 6) + z 2 P 2 (cos 0) + . . . . 



It will easily be verified that 

P l (cos 6) cos 6, 
P 2 (cos0) = 
The functions are orthogonal: 



r 

Jo 



P,(cos 0) p r (cos 0) sin 0d6 = 0. 



The importance of the Legendre functions is that, in spherical 
polar coordinates (r, 0, <f>), the general solution of an equation 
of the type 

= 0, (24) 



which is a function of r, alone, is 

P,(cos0)/(r), 



20 DIFFERENTIAL EQUATIONS OF WAVE MECHANICS 

where /(/*) satisfies 



r dr 



(25) 



The reader will easily verify this for / = (),/ = I. 

To obtain the most general solution which is a function also 
of <f> one has to introduce the associated Legendre polynomials, 
P; M (cos 6). where u has the values 0, 1, 2, ...,/. Some values are 



It 





I 


" "-" ' " - 1 
2 





1 


cosfl 


^cos^-i 


1 




shift 


JJcos^sin 6 


> 

- 






:\*\\\*6 

._... , . 



The general solution of (24) is 



where /still satisfies (25), and u has 2/4- 1 integral values from 
-/to 4-/. 



CHAPTER II 
THE WAVE EQUATION OF SCHROD1NGER 

1. THE WAVE FUNCTION 

Wave mechanics is a system of equations which determines the 
behaviour of the fundamental particles of physics, the electron, 
the proton, the neutron, and their interaction with radiation. 
In its present form it appears adequate to describe the behaviour 
of the electrons outside the atomic nucleus sufficiently accur- 
ately to account for the known facts of spectroscopy , chemistry, 
and so on. Within the nucleus it has had some success, notably 
in giving an explanation of a-decay; but at the time of writing 
not sufficient is known about the forces between the constituents 
of the nucleus for a forecast to be made of its ultimate success 
in this field. 

In this chapter we shall limit ourselves to the application of 
wave mechanics to electrons. The theory will be based on a 
single experimental fact, the diffraction of electron beams. 
This was first discovered by Davisson and Germerf and by 
G. P. Thomson, J and has now become a useful technique of 
applied physics. Briefly the experimental facts are as follows. 
When a beam of electrons passes through a crystalline sub- 
stance, such as a metal foil, the beam is scattered by the sub- 
stance in exactly the same way as a beam of X-rays is scattered. 
Thus diffraction rings are produced by a beam which has 
penetrated a poly crystalline foil; and from a single crystal a 
beam of electrons is reflected according to the Bragg law. The 
beam of electrons thus behaves as though it were a beam of 
waves, and the wavelength can be determined; it is related to 

f C. Davisson and L. H. Conner, Phys. Rev. xxx, 707, 1927. 
{ G. P. Thomson, Proc. Roy. Soc. A, cxvn, <H) t 1928. 



22 THK WAVE EQUATION OF SCHRODINOEB 

the momentum p of each electron by the equation 

A = /?/?>. (1) 

where h is Planck's constant,! 

Tn this book we prefer to treat (1) as given by experiment, 
and thus as the observed fact on which the whole theory of 
wave mechanics must be based. It was predicted, however, by 
Louis de BroglieJ before it was discovered experimentally. 
A simplified version of his argument is as follows: 

If there is some relationship between the momentum vector 
p of the particle and some property of a train of waves, it must 
be obtained by equating p to some vector which describes the 
wave motion. Now a train of waves, travelling in the direction 
defined by the direction cosines (l,m,n), may be written 

sin {k(lx -f my + nz) a>t}. 

k is here the wave number, or 2?r multiplied by the reciprocal 
of the wavelength. Since k(lx + my + nz) is a scalar, and (#, y, z) 
is a vector, it follows that the quantity with components 

(kl, km, kn) 

is itself a vector. This we call the wave vector, and denote it 
by k. Thufl if a correspondence of the type envisaged exists 
between the momentum of a particle and the wavelength of a 
wave, it must be of the form 

p = const, k. 

That the constant should be Planck's constant divided by 2-n 
was suggested by the existence of a similar relationship for 
light quanta, where the momentum was known to be hv/c, in 
other words hk/^-n-. 

Returning now to beams of electrons, one can define more 
precisely the observed behaviour as follows. In any problem in 

t For experimental proof that beams of particles of atomic mass behave 
in the same way, cf. F. Ktiauer and O. Stern, Z. Phys. LIII, 786, 1929, or 
1. Estermann and O. Stern, Z. Phys. LXI, 115, 1930. 

t L. de Hroglic, Phil. Mag. XI.VH, 44, 1924; Ann. Phys,, Paris, in, 22, 1923. 



THE WAVE FUNCTION 23 

which it is desired to calculate the path of a beam of electrons, 
its scattering by atoms or crystals or its bending by electric or 
magnetic fields, one has to postulate the presence of a wave, 
and calculate the wave amplitude everywhere. Then the 
density of electrons at any point will be proportional to the 
intensity of the wave at this point. All this is merely the 
expression of an experimental fact, the diffraction of electrons 
by crystals. 

We denote the amplitude of this wave by T. Since no way 
of measuring its amplitude exists except through the property 
that the intensity is proportional to the density of electrons, 
it is reasonable to choose our 'units' so that the square of the 
modulus of X F is equal to the density of electrons; thus 



where N is the number of electrons per unit volume. 

The quantity Y, known as the wave function, is a complex 
quantity 



The square of the modulus is thus defined by 



The asterisk is used throughout this book to denote the complex 
conjugate of a complex quantity. Thus if 



then *F*=/-tV. 

It is often a stumbling-block to the beginner in this subject 
that a physical quantity, the wave function, should be repre- 
sented by a complex quantity. The reason is as follows. We 
know a priori nothing about the wave function, but we should 
expect, by analogy with the case of light waves, that the type 
af expression which in other wave systems represents the 
energy density would in this case give the particle density. 



24 THE WAVE EQUATION OF SCHRODINGER 

But the energy density in any wave system is always given by 
the sum of the squares of two independent quantities whose 
magnitudes define the state of the wave. For a light wave 
these quantities are E and H and the energy density is 
(jf? 2 -f// 2 )/877. For elastic waves they are the displacement and 
velocity of the medium. Thus, for the waves associated with 
electrons or other material particles, it is reasonable to assume 
that the state of the wave at any point is defined by two 
quantities / and g, and it is convenient to combine them into a 
single function T by means of (2). 

It may be noted that Maxwell's equations for the electro- 
magnetic field in free space may be treated in the same way; 
the equations are 

._ 9H ... 8E 

- c curd E = , c curl H = ~~, 
ct <jt 

and if T is written for E -f ?'H, both equations may be combined 
in the single equation 

.ff 

i-Tr- 

ct 

It will be convenient at this stage to make a further assump- 
tion about the form of a plane wave. A plane wave travelling, 
say, along the x-axis has, for any type of wave motion, the 
form 

A sin (lex wt + e), 

where k is the wave number, o>/27r the frequency, and e a phase. 
In a plane polarised light wave E and H are in phase; thus the 
energy density is proportional to 



and fluctuates with time at any point. There is no reason to 
think that any such fluctuation occurs in the wave associated 
with an electron; it would in fact be difficult to understand 
what physical significance could be ascribed to a rapid fluctua- 
tion of the probability that an electron would be found at a 



SCHRODINGER'S EQUATION 25 

certain point. It is therefore reasonable to suppose that/ and g 
are 90 out of phase, so that, A being a constant or a slowly 
varying function of x t 

/= A cos(kx wt), 
y A sin (kx a>t), 
and |Y| 2 =/ 2 + 2 = .4 2 . 

With this convention | T | 2 keeps a steady value independent 
of time. Making use of the complex function T we see that a 
plane wave moving from left to right is represented by 



We shall represent a wave going in the opposite direction by 



In the remainder of this book we shall follow the accepted 
convention and use always the complex wave function *F, and 
shall not refer again to the real and imaginary parts, / and g. 

2. SCHKODINGER'S EQUATION 

We shall now write down Schrodinger's wave equation in 
the form appropriate to a beam of electrons, each of total 
energy W, moving in an electrostatic field. 

We represent by V(x, y, z) the potential energy of an electron 
in this field; thus in a uniform electrostatic field E in the z 
direction, for example, we should have 



for an electron in the field of a nucleus of charge Ze, 



where r = <J(x 2 -f y 2 -f z 2 ) is the distance from the nucleus. W , the 
total energy, is equal to the kinetic energy at the point where 
we arbitrarily choose V(x, y y z) to be zero, at z = for the first 
case and r = oo for the second. Then at the point (#,t/, z) the 



26 THE WAVE EQUATION OF SCHBODINGEK 

kinetic energy of one of the particles is W V(x,y,z). There- 
fore the experimental relation (1) shows that the wavelength A 
of the accompanying Wave is 

A = A/V{2w(JF-F)}. (3) 

Now it is clear from the considerations of Chapter I, 2 that 
for motion in one dimension a function T which oscillates in 
space with constant wavelength A satisfies the equation 



and that the generalisation to three dimensions is 



The simplest assumption that we can make is that the same 
equation is satisfied where A varies from point to point. Thus 
substituting from (3) we obtain 

( W - V) T = 0. (4) 

This is Schrodinger's equation for the wave function *F. 

It is convenient to introduce the symbol ft to denote h/2n. 
With this notation the wave equation becomes 



-i~ = 0. 

n* 

Certain results of this equation must be verified before it can 
be regarded as satisfactory. It must be shown: 

(i) That it makes correct predictions about the bending of 
beams in electric and magnetic fields, where the classical 
Newtonian mechanics is known to give correct results. 

(ii) That it predicts that the total current in a steady beam 
does not vary from point to point, so that the equation does 
not predict the creation or annihilation of particles. 

We shall attend to these points in turn. 



BENDING OF A BEAM OF ELECTRONS 27 

3. BENDING OF A BEAM OF ELECTRONS 

According to Newtonian mechanics, the bending of a beam by 
an electric field can be calculated as follows. The force on each 
particle normal to the beam is dV/dn, where 8/dn denotes 
differentiation normal to the beam. The radius of curvature R 
of the orbit of each particle is obtained by equating this to the 
centrifugal force, mv 2 /R, which may be written 2(WV)/R. 
Thus 

1 2V I 

W-V). (5) 

We wish to show that the same formula for the bending of a 
beam of electrons can be obtained by means of wave mechanics. 
We have already shown that if A, the wavelength, is a function 
of position, the radius of curvature of the beam is given by (cf. 
Chap. I, 2): 

R Xdn 

Since A = h/J{2m(W V)}, we see that formula (5) follows. 
Thus, in so far as the effect of an electric field is concerned, 
classical and quantum mechanics give the same result. 

A similar result may be obtained for a magnetic field using 
(13), but the proof will not be reproduced here. 

4. SOLUTIONS OF SCHRODINGER'S EQUATION AND THE 
CONSERVATION OF THE NUMBER OF PARTICLES 

In the next chapter it will be shown that the frequency v of 
an electron wave is related to the total energy W of the electron 
which it represents by the equation 



Thus if ifj(x,y,z) is any solution of (4) describing the behaviour 
of a beam of particles each of energy W y the full form of the 
wave function is 

v - WJh. (6) 



28 THE WAVE EQUATION OF SCHROD1NGER 

The use of the negative sign in the exponential is simply a con- 
vention. As long as we are dealing with steady beams of 
particles all having the same energy, the time will enter into 
the wave function through a factor e~ 2niwi/h as in (6). It will 
thus simplify all formulae dealing with such beams if we write 
down the functions iff and not the functions X F; a plane wave 
going from left to right, for instance, will be written = e> kjc . 

With this simplification, certain examples will be considered 
which illustrate the novel features of wave mechanics and also 
verify the conservation of particles. We shall begin by con- 
sidering the motion along the .r-axis of a beam of electrons 
of infinite width in a field of potential energy V(x). The 
Schrodinger equation then becomes 



In the absence of a field (V = 0), the solution representing a 
beam moving from left to right is 



- Ae ikx (k 2 = Z 
which represents A* electrons per unit volume, or 
A 2 v (}>mv 2 = W) 



crossing unit area per unit time. In the presence of a field we 
may distinguish two cases: 

(i) V(x) varies slowly from point to point. An approximate 
solution may then be obtained by the W.K.B. method (Chap. I, 
1) and is 



where A is a constant, and the lower limit of the integral is 
arbitrary. The form of the solution is illustrated in Fig. 8 for 
the case where V = eEx, and thus for an electron accelerated 
by an electric field. It will be seen that as the electron is 
accelerated, so that the wavelength shortens, the amplitude 



THE CONSERVATION OF PARTICLES 



29 



also decreases. The number of electrons per unit volume is, 
by (7), 

J2 

IT! 2 = - 

1 ' (W-V)* 

But W V is the kinetic energy, so that | T | 2 is inversely pro- 
portional to v, the velocity of the particles at the point con- 



V(x) 




Fig. 8. Potential energy function and wave function $ for an 
electron accelerated by a field. 

sidered. Thus v\ T | 2 , the number of particles crossing unit area 
per unit time, is the same at all points of the beam. The 
conservation of particles is thus verified. 

If V(x) is a slowly varying function of x, then the predictions 
made by wave mechanics are the same as those of classical 
mechanics; the electrons are accelerated by the field, and none 
of them is reflected. Here, as in the bent beam treated in 3, 
wave mechanics makes no new predictions. If, however, V(x) 
varies significantly in a distance small compared with the wave- 
length, the predictions of wave mechanics are entirely different 
from those of classical mechanics. This case will now be treated. 

(ii) We may consider an extreme case, a 'potential jump', or 
in other words a plane perpendicular to the x-axis at which the 
potential energy function V(x) changes discontinuously. This 



30 THE WAVE EQUATION OF SCHROD1NOER 

example is introduced in order to illustrate the principles of 
wave mechanics; no case of a discontinuous potential jump 
exists in nature. The example closest to that discussed here is 
perhaps the rapid change in the potential energy function 
which exists at the surface of a metal (cf. Chap. V, 7). 
We set then for V(x) 

V(x) = #<(), 

= F O x>o, 

and consider a stream of particles each of kinetic energy 
W(W > V ) incident on the potential jump from the left. At the 
potential jump there is a sudden change of wavelength; the 
wave-number changes from 



to V = 

Therefore according to the arguments of Chapter I, 2, the 
wave must be partially transmitted and partially reflected, [n 
order to calculate how much is reflected, and how much trans- 
mitted, it is necessary to know the boundary conditions satis- 
fied by the wave function. These are that ^ and difj/dx are 
continuous. This may easily be seen, since 



dx 

and, although the integrand is discontinuous, the integral (which 
represents the area under a curve) must be continuous. 

With these boundary conditions, the analysis of Chapter I, 
2, may be applied as it stands. With an incident wave of 
amplitude unity (e ikx ), the amplitude of the reflected wave is 
(k k')l(k + &'), and that of the transmitted wave is 2k/(k + k'). 
The numbers of particles incident, reflected and transmitted 
per unit area per unit time are 

v incident, 

v(k- k')*l(k + k' ) 2 reflected, 
and 4t/2/(fc + &')2 transmitted. 



THE CURRENT VECTOR 3i 

The proportion R reflected is thus 



and the proportion T transmitted is 
T = 



It will easily be verified that T + R is equal to unity. It is thus 
verified again that the wave equation chosen is compatible 
with the conservation of charge. 

The prediction made by wave mechanics, that some of the 
particles are transmitted and some reflected, is of course fully 
at variance with classical mechanics, according to which they 
would all be transmitted. We see then that wave mechanics is 
unable to make a definite statement about the behaviour of an 
electron incident on a potential jump; it only allows a calcula- 
tion of the average numbers transmitted and reflected, or in 
other words the probability that a given electron is transmitted 
or reflected. This inability to make exact predictions about 
the behaviour of individual particles is a general property of 
wave mechanics. 

5. THE CURRENT VECTOR 

The student will easily verify that, for a wave function of 
the type 



the number of electrons v(\A | 2 | B\ 2 ) crossing unit area per 
unit time may be written 



2mt 



A general proof that this quantity is independent of x is of 
interest. Since tft and 0* satisfy the equations 



d 



32 THE WAVE EQUATION OF SCHRODINGER 

we have, multiplying the first equation by 0* and the second 
by if/ and subtracting, 



In other words 



It follows that the current is independent of x. If this were not 
a consequence of the wave equation, the equation would lead 
to incorrect results. 

In three dimensions the vector 



represents the number of electrons per unit time crossing unit 
area perpendicular to itself, and is known as the current vector. 

Exercise 

Prove the theorem equivalent to (8), that 

divj - 0. 

0. THE TUNNEL EFFECT 

We have not yet considered the description in wave mechan- 
ics of a beam of electrons entering a field which opposes their 
motion and eventually stops them and turns them back. To 
describe what happens, we shall consider a beam of electrons 
moving from left to right along the #-axis and at x = entering 
a field E. The potential energy of an electron is then given by 

V(x) = eEx; 

the kinetic energy is WV, so the electrons are stopped and 
turned back when x = WjeE. 

To describe the behaviour of the electrons according to wave 
mechanics, we have to solve the Schrodinger equation 



THE TUNNEL EFFECT 33 

for this form of V(x). Consider first the form of the solution 
to the right of the point where x W/eE. W V is then 
negative, and the arguments of Chapter I, 1 show that there 
are two solutions, one of which increases with increasing x and 
one of which decreases. The solution which represents the 
physical state of affairs is the one which decreases; the other 
solution would represent a rapidly increasing density of 
particles beyond W/eE, which is absurd. We choose then the 
decreasing solution; this i&, of course, a real function of #. 

To the left of x = W/eE, then, we have an oscillating solution 
</r(x), which, since it fits onto the real solution to the right, will 
be real too. Thus the complete wave function with the time 
factor 



represents a standing wave, that is to say, an incident wave 
and a reflected wave having equal amplitudes. 

The description given by wave mechanics of the phenomenon 
is not very different from that given by classical mechanics; all 
the electrons are reflected. The only difference is that they are 
not all reflected exactly at the point where x = W/Ee; some of 
them penetrate a little further. There is thus a finite if small 
probability of finding an electron at any distance beyond the 
point where x = W/Ee. 

This fact has important consequences; it gives to an electron 
a finite probability of penetrating through what is called in 
quantum mechanics a 'potential barrier'. A potential barrier 
is illustrated in Fig. 96, formed by two fields; a field E opposing 
the electron's motion from x to x = a, and a field E in the 
opposite direction from x = a to x 2a. The potential energy of 
the electron in this field is 

V(x) = Eex Q<x<a, 

V(x) = Ee(2a -x) a<x<2a. 

In general a potential barrier is a region in which V(x) > W 
sandwiched between two regions in which W > V(x). According 



34 THE WAVE EQUATION OF SCHBODINGEB 



Ecx 




V(x) 




Fig. 9. (a) Electrons reflected by a field, (b) A potential barrier 
due to two fields as described. V is the potential energy of an 
electron and $ the wave function. 



THE TUNNEL EFFECT 35 

to classical mechanics, particles incident on such a region will 
all be reflected; according to wave mechanics a certain pro- 
portion of the particles will penetrate the barrier and come out 
the other side. This prediction of wave mechanics is known in 
the literature as the 'tunnel effect'. 

If iff j, i/r# are the amplitudes of the wave function at the two 
extremities of the barrier, the chance of penetration P is given 
approximatelv by 

f-i^/^.1 1 - () 

In many practical cases P can be calculated by the W.K.B. 
method. Neglecting the factor outside the exponential in (7), 
which in general changes little in comparison with the expon- 
ential factor, we see from (9) that 

(10) 

the integration being from one extremity of the barrier to 
the other. 

Some physical phenomena in which the tunnel effect is 
important are: 

(a) The escape of a-particles from a radioactive nucleus 
(Chap. VII, 3). 

(6) The escape of electrons from a metal under the action of 
a strong field (Chap. V, 7). 

(c) The passage of electric current between two metals 
separated by an oxide layer. 

The discussion of waves with imaginary refractive index 
given in Chapter I, 2, is relevant to the phenomena con- 
sidered here. 

It is important to estimate how thick a barrier electrons can 
in fact penetrate. Let us suppose that two metal wires are 
separated by an air gap or an oxide layer of thickness a. If a 
potential difference of, say, half a volt is applied across the gap, 
a certain current will pass. Let us suppose that the area of the 
contact is 1 sq. mm. A metal contains about 1C 23 free electrons 



36 THE WAVE EQUATION OF SCHRODINGER 

per c.c., and they move with a speed of about 10 8 cm. /sec., so 
that 10 29 will impinge on each side of the gap each second. If 
half a volt is applied, about one-tenth of the mean kinetic 
energy of the electrons (cf. Chap. VI, 7), we may suppose that 
one-tenth of the electrons have not energy enough to pass 
through the gap against the field. Thus the number of electrons 
passing through the gap in the direction of the field is, each 
second, about 

10 28 P, 

where P is the quantity defined by (9). A current of one 
ampere corresponds to c. 10 19 electrons per sec., so we see 
that a current of this magnitude will flow if P = 10~ 9 , while 
a milliampere will flow if P = 10~ 12 . Taking for the height <f> 
of the barrier a quantity of the order of the work function, 



we have, since 

exp{- lV(2m<) a/h} = P, 



pj cm * 

Thus a gap of 10~ 7 cm. would give one ampere, 1-4 x 10~ 7 cm. 
a current of one milliampere. It will be seen that the current 
drops very rapidly as the thickness is increased, and barriers of 
c. 2 x 10~ 7 cm. are practically opaque. 

7. SCATTERING OF BEAMS OF PARTICLES BY ATOMS 

An important class of question which wave mechanics can 
solve is that of the scattering of beams of particles by atoms 
or nuclei. The problem can be put as follows. A substance, for 
example a gas, contains N scattering centres (atoms, nuclei) 
per unit volume. A particle (electron, proton, a-particle) moves 
through the substance. What is the probability, per length x 
of its path, that the particle is scattered through an angle 6 
into the solid angle dco ? We denote this probability by 

NI(6)xda>. 



SCATTERING OF BEAMS OF PARTlCLKS 37 

It is clear that 1(6) has the dimensions of an area. The integral 

iir sili 



^ (l(d)da>= f 



giv r es the effective total cross-section of the atom or other 
centre for the type of collision in question. In other words, 
NA is the chance of a collision per unit length of path. 

By the methods developed in this chapter it is possible only 
to calculate the scattering of particles by a centre of force, for 
example the scattering of a-particles by the field of a heavy 
nucleus. The scattering of electrons by atoms is a many-body 
problem, involving the possibility of transfer of energy between 
the incident electron and the electrons of the atoms. It is, 
however, possible to consider approximately elastic collisions 
(those in which the electron loses no energy) by representing 
the atom as a centre of force. It will be shown in Chapter IV, 
6 how this force may be calculated. We denote by V(r) the 
potential energy of an electron acted on by this force. 

The problem, then, is to calculate 1(0) for a stream of 
particles incident on a region, at a distance r from the centre 
of which the potential energy of any one particle is V(r). One 
has therefore to find a solution of Sehrodinger's equation 



which represents an incident plane wave and a scattered wave. 
Such a solution must have the form, for large r, 

p ikr 
if, e tkx + ?--/($). 

Here the first term represents the incident wave moving from 
left to right along the #-axis, the second the scattered wave 
which must fall off inversely as ;. The solution represents a 
state of affairs in which v particles cross unit area per unit time in 
the incident beam; in the scattered beam there are r~ 2 \f(B)\ 2 
particles per unit volume at a distance r from the scattering 



38 THE WAVE EQUATION OF SCHRODINGER 

centre. Thus v\f(0)\ 2 dw cross an area r 2 doj per unit time; 
hence 



One of the most important cases is that in which V = ZZ'e z lr, 
corresponding to the scattering of particles of charge Z'e(Z' = 2 
for a-particles, Z' = 1 for electrons) by a bare nucleus of 
charge Ze. Here an exact analysis! shows that 

__ I 



It is remarkable that for this case and for this case alone 
wave mechanics yields the same formula as classical mechanics. 
This formula was in fact derived from classical mechanics by 
DarwinJ and used by Rutherford to interpret his experiments 
on the scattering of a-particles by metal foils which established 
the nuclear model of the atom. 

For a more general field V(r), it may be shown that each 
element of volume dxdydz scatters a wavelet of amplitude, at a 
distance R from the element, 

(r)dxdydz*t, (11) 

where ifj represents the amplitude of the whole wave at that 
point. This may be shown in two ways. The rigorous method 
is to make use of a theorem known as Green's theorem (outside 
the scope of this book); the proof is given by Mott and Massey. 
A more elementary method is as follows. Consider a beam of 
electrons moving along the #-axis and incident on a region of 
sheet-like form in which the potential is defined by 

V = x < 0, 
= V Q 0<x<a, 
= a<x, 

where a is some small distance. We shall suppose that W > V . 

f \V. Gordon, Z. Phys. XLVIII, 180, 1928; Mott and Massey, chap. HI. 
t C. G, Darwin, Phil. Mag. xxvii, 499, 1914. 
Mott and Massey, chap. vi. 



SCATTERING OF BEAMS OF PARTICLES 39 

We may calculate the amplitude reflected as follows. We set 



= Bef k ' x + Ce~ ik ' x 

= De ikx a < x, 

where k 2 - k' 2 =* 2mVJH*. 

Putting in the boundary conditions that </r and di/f/efa; are 
continuous at x = and x a, and making also the assumption 
that k'a<^I, we find 



Now our problem is to find the amplitude of a wavelet 
scattered by a volume element dxdydz in which the potential 
is V Q . Let this be (xdxdydz/R. A surface element of the sheet of 
area IdS will then scatter a wavelet aadS/R. These wavelets 
add up to give a reflected wave of amplitude, at a point distant 
x from the sheet, 

/QO e ikK 

oca --- 27Tzdz, 
Jo M 

where R 2 = z 2 -4- x 2 \ on integration this gives, for x < 0, 



This has to be equated to Ae~ ikx . Thus 



whence <x = (P-' 2 )/47r = 2<rrmV /h z , 

which is what we set out to prove. 

The result (11) may now be used to calculate the scattering 
of electron waves by a centre of force, if W is everywhere great 
compared with F. Under these conditions the form of the 
wave cannot be greatly perturbed within the atom, so that in 
calculating the scattered wavelets one can assume that $ a * 



40 



THE WAVE EQUATION OF SCHRODINGER 



any point is given by e ikx . This approximate method of obtain- 
ing the scattered amplitude f(6) is known as the Bornf approxi- 
mation. In fact, of course, near the centre of an atom V(r) 
always becomes large; nevertheless, the Born approximation 
does give fair results for electrons of high energy. Making use 
of Bom's approximation, the amplitude of the wave scattered 
through an angle may be calculated as follows. Let OXZ be 




Fig. 10. 

the plane normal to the bisector of the angle between the inci- 
dent and reflected rays (Fig. 10). Then all the wavelets scat- 
tered from any plane parallel to OXZ will be in phase with 
each other, and the wavelets scattered from two such planes 
distant y from each other will have a phase difference py, 
where, as may easily be verified, 



k = 27T/A. 

The resultant, therefore, of all scattered wavelets will be, at 
large distances R from the atom, f(0)/R, where 



t M. Born. Z. Phys. xxxvn, 863, 1926; Z. Phys. xxxvm, 803, 1926. The 
former paper is the first in which the probability interpretation of the wave 
function is introduced explicitly. 



SCATTERING OF BEAMS OF PARTICLES 41 

If we take spherical polar coordinates (r, 6', </>') such that 
y = rcos 0', this becomes 



2-TrWl 

o o 



Writing cos #' = 2, sin#'d0' = dz, we find finally on inte- 
grating over z 



The integration may be carried out for various forms of 
V(r). 

Exercises 

(1) Carry out the integration in (12) for the screened 
Coulomb field 

Ze 2 
V(r) = ---exp(-gr). 

Show that as ry->0, the Rutherford scattering formula is 
obtained. 

(2) If V(r) is the potential energy function due to a nucleus 
of charge Ze and a negative charge distribution of charge 
density ep(r), show that (12) can be put in the form 



where F(B) = TT f p(r) ~^ r*dr. 

Jo K 

Interpret this as showing that each element dp of charge 
scatters according to the Rutherford law. 

(3) From (12) show that the scattering is spherically sym- 
metrical if the radius of the atom is small compared with 
A/27T, where A is the wavelength of the incident wave. Show 
that this is true, in general, of the exact solution of the wave 
equation. 



42 THE WAVE EQUATION OF SCHKODINGER 

(4) Find the scattering due to a hard sphere of radius a, on 
the surface of which $ may be assumed to vanish; you may 
take a<^A/27r. Show that the total cross-section is 4?ra 2 . 

8. ELECTRONS IN A MAGNETIC FIELD 

The Schrodinger equation for an electron moving in an 
electromagnetic field isf 

= 0, (13) 

where A is the vector potential of the field, defined by 

H = curl A. 
For a uniform field H along the .c-axis, the vector potential is 



so that the equation reduces to 



Exercise 

Using (13), prove (8) verifying the conservation of charge. 



t For the proof, see, for example, N. F. Mott and I. N. Sneddon, Wave 
Mechanics and its Applications, 1948, p. 39. 



CHAPTER III 

WAVE GROUPS AND THE UNCERTAINTY 
PRINCIPLE 

1. THE FREQUENCY OF ELECTRON WAVES 

In the last chapter we have considered the description in terms 
of wave mechanics of beams of electrons, each electron of which 
has the same energy W. We have introduced a wave function 
T of the form 

Y0(s, 0,2)6-'", (1) 

where i/j satisfies the equation 

*")?*) 
?(H^-F)0 = 0, (2) 



and have interpreted the solution by saying that | T(x,j/, z\t) | 2 
is the average density of electrons in the beam at the point 
(x y y, z), or that | V \ 2 dxdydz is the probability that an electron 
will be found at any moment in the volume element dxdydz. 
We now show how to apply wave mechanics to a more general 
case than that of steady beams, namely, to a state of affairs 
where the density varies with the time. At the same time we 
shall introduce an expression for the frequency v( o>/27r) of 
electron waves. 

We consider first the following idealised experiment. Sup- 
pose that a beam of particles each having velocity v is incident 
on a screen, in which there is a hole which can be closed by a 
shutter (Fig. 11). The shutter is closed initially, then opened 
for a time t Q and then closed again. Then a beam of length vt Q 
will pass through the hole and move forward with velocity v. 
According to the concepts of wave mechanics, however, we 
must describe the whole phenomenon in terms of the wave 
function T. A continuous train of waves falls on the screen; 



44 



WAVE GROUPS 



vt n 



when the shutter is opened a train of waves of limited length, 
that is to say a wave group or wave packet, is allowed to pass 

through. As usual, we must set 
the intensity of the wave at any 
point equal to the density of elec- 
^ . ^ trons there. The wave group must 

* \ //^ trave l wit h t* 16 g rou P velocity of 

* the waves; thus if wave mechanics 

is to give a correct description 
of the observed phenomena, the 
group velocity of the waves must 
be equal to the actual velocity of 
the particles that they represent. 
The group velocity in any type of wave motion is (cf. Chap. I, 
3) dw/dk, where CD ( = ZTTV) is defined by (1). We must thus set 



Ffc. 11. 



da) 
~dk 



v. 



(3) 



Up to the present we have not ascribed any physical meaning 
to the frequency v of electron waves. By integrating (3), how- 
ever, we may find an expression for to and hence v. For, by 
equation (I) of Chapter II, we see that 



k = mv/H. 

dw Hk 
dk m' 



Thus (3) becomes 

On integrating we find 



2 /H -h const. 
Thus for a freely moving particle we may set 

ft to = hv = kinetic energy -f constant. 

We next have to consider the value of this constant. It will 
be realised that a steady beam of particles moving through a 
field of force must be represented by a wave with frequency 



FREQUENCY OK ELECTRON WAVES 46 

the same at all points. It is natural to take a point where the 
potential energy is zero and to define hv as the kinetic energy 
there. We see therefore that 

Ha> = hv = W , (4) 

where W is the total energy of each electron. It must be 
realised, however, that the point where the potential energy 
vanishes is quite arbitrary, and so the total energy of a particle 
in a field of force contains an arbitrary constant. 

It is rather surprising that the frequency of these electron 
waves should also contain an arbitrary constant; it suggests 
that, though the equations of wave mechanics are correct in 
their description of how matter actually behaves, these waves 
have not the same sort of physical reality as sound or electro- 
magnetic waves. This view will be confirmed by the considera- 
tions of Chapter V. 

2. THE WAVE EQUATION FOR NON-STATIONARY PHENOMENA 

The wave equation (2) applies to steady beams; it will not 
apply to wave groups such as that shown in Fig. 11. The 
equation that we require must contain terms in d*j8t, so that 
it may be used to calculate the future motion of a wave group 
when its initial form is given. Moreover it must be of the first 
order in the time; that is to say it must contain terms in 
3T/& but not in d^/dt 2 . This is because, as we have seen in 
Chapter II, the complex function V contains two real terms / 
and g which are analogous to the displacement and the velocity 
of waves on a string, or to E and H in electromagnetic theory; 
thus a knowledge of T alone at a given time, without a know- 
ledge of 3T/5/, should suffice for the calculation of its value at 
all subsequent times. Only if the equation is of the first order 
will this initial condition be enough. 

The required equation can be obtained by eliminating W 
from (2). From (4) we have for the typical wave function 
describing electrons of energy W 

T(:r, y, z; t) = $(x. y> z) e~ im/ *. (5) 



46 WAVE GROUPS 

Differentiating this equation, we obtain 



-. 
dt H 

Substituting in (2), we find 



.. 

i ft! 2m 

This is the required equation. Its most general solution is 
made up by superimposing solutions of the type (5): 



rr 



As will be shown in Chapter VT, such a solution represents a 
state of affairs in which the energy of the electron is not known, 
but the probability that it has the value W is | A w | 2 . 

Exercise 

Make use of (6) to verify the conservation of number of 
particles, namely, to prove that 



the integral being over all space. 

Notation 

It is often convenient to write H for the operator, 



2m 
so that the Schrodinger equation (6) becomes 



i dt 

and the equation (2) 

(H-W)f-O. 



DESCRIPTION OF SINGLE PARTICLE 47 

3. THE DESCRIPTION IN WAVE MECHANICS 
OF A SINGLE PARTICLE 

The wave group described in 1 and illustrated in Fig. 11 
describes a number of electrons; the integral 

\*dxdydz (7) 

gives the average or probable number of electrons passing 
through the hole while the shutter is open. It need not be an 
integer; and it may be less or greater than unity. We might, 
however, imagine the shutter open just long enough to let, on 
the average, one electron pass through; then the integral (7) 
will be set equal to unity, and the volume occupied by the 
wave group, the shaded area in Fig. 11, represents the space 
where, as a result of the experimental arrangement illustrated, 
the electron may be. 

This arrangement with a shutter may be thought of as just 
a way of obtaining approximate information about the position 
and velocity of a particle. Many other devices may be imagined. 
Given any such device we may formulate as follows the way 
in which wave mechanics must be used to make predictions 
about the future position and velocity of the particle. We con- 
fine our description to movement in one dimension, though it 
may at once be generalised to three. Suppose that measure- 
ments are made, at a given instant of time, of the position and 
momentum of th6 particle. Suppose that the results of these 
measurements are that the position is at X Q with a probable 
error | A#; and that the momentum is p Q with a probable error 
\tp. Now if Ax AJP is not too small, the result of these measure- 
ments can be described by a wave function having the form of 
a wave group. The chance that the particle is between the 
points x,x + dx as the result of our measurement may be written 
const, exp {(x - x ) 2 /(|Ax) 2 } dx. 

This by the rules of wave mechanics, is equal to \*\ 2 dx\ we 
thus set at t = 

T = const. e ik ** exp {(x - z ) 2 /2(Aa;) 2 }, (8) 



48 WAVE GROUPS 

where k = p /ft. Such a wave function, then, represents a 
particle at the required position, and with momentum approxi- 
mately equal to p Q . But (8) can be expanded, as shown in 
Chapter I, 3, in the form 

if/ = \A(k)exp{ik(x x Q )}dk y 

where A(k) = const, exp { - (k - i ) 2 (|Ax) 2 /2}. 

The wave group is thus made up of simple harmonic waves 
with & in a range about & of A&, where 



Afc = 4,/2/Ax, 

as may indeed be seen without mathematical development 
from the considerations of Chapter I, 3. The wave group 
thus describes particles with momenta p lying in the range 
determined by | A(k) | 2 , and thus between p Q ^A^, where 



To a good enough approximation, this may be written 

ApAz~A. (9) 

Equation (9) states the uncertainty principle of Heisenberg. 
If measurements are made so that Aj9 Aa: is greater than h, it is 
still possible to imagine a wave group set up, similar to a wave 
group of white light, containing waves having a range of fre- 
quencies; but if A# A# is less than A, it is impossible to set up a 
wave group to represent the results of the measurements. We 
are thus driven to the conclusion that 

either measurements for which &x&p<h are impossible in 
the nature of things, 

or it is impossible to describe the motion of an electron by 
means of wave mechanics. 

The facts of electron diffraction seem to rule out the second 
alternative; we are thus driven to believe that there is in fact 
a limiting accuracy of all measurement. We shall come back 
to this point in the next section. 



DESCRIPTION OF SINGLE PARTICLE 



49 



Once a wave group has been set tip describing the results of 
the initial measurement, the wave equation (6), being linear in 
d/8t, will predict its form at any future time. Thus at any 
future time it is possible to state the probability |MP*(a?, t) \ 2 dx 
that the particle will be found between x and x + dx. This then 
is the type of prediction that wave mechanics enables one to 
make: given certain initial measurements, made with a certain 
probable error, one can predict the chance that at any future 





Classical mechanics 



Wave mechanics 



Fig. 12. Showing the contrast between the classical and wave- 
mechanical method of prediction. In the classical method a measure- 
ment shows that the particle is in the volume AH moving within 
the directions shown by the arrows. By considering all the orbits 
such as PQ consistent with this original measurement, one arrives 
at the conclusion that after time t the particle will he within the 
volume CD. The wave-mechanical treatment pictures a wave packet 
moving from AR to CD, passing through the intermediate posi- 
tion EF. 

time a particle (or system of particles) will be found at a given 
point. In this, wave mechanics is similar to classical mechanics; 
but classical mechanics proceeds by calculating the system of 
orbits which are consistent with the original measurement; 
these are absent in wave mechanics. The difference between 
the two methods is illustrated in Fig. 1 2. 

In fields which vary slowly with the distance it may be 
shownf that the wave group of wave mechanics follows the 

f Cf., for example, Mott and Sneddon, chap. i. 



50 WAVE GROUPS 

same path as the particles of classical mechanics; it is only 
when we have to deal with fields varying in a distance com- 
parable with the wavelength of an electron (c. 10~ 8 cm.) that 
the two systems give different results, as in the diffraction or 
scattering of electrons by atoms. 

4. THE UNCERTAINTY PRINCIPLE 

We have seen that, if wave mechanics is valid, measure- 
ments must be impossible unless 



It will be of interest to estimate the magnitude of these quanti- 
ties. If we write p = mv, 



~ 7 c.g.s. units for an electron. 

Thus if Ax is 1 cm., At; ~ 7 cm. /sec., which is of order one part in 
10 8 of the velocity of electron in an atom. If, however, Ax is of 
the order of the size of an atom (10~ 8 cm.), Av/v~ 1. 

It is of great interest to examine the hypothetical experi- 
ments by which we could determine position and momentum 
simultaneously, and to show that they do in fact yield an 
uncertainty of the predicted amount. The most famous of 
these demonstrations is the 'gamma-ray microscope' first dis- 
cussed by Heisenberg. The argument put forward is as follows. 
A beam of electrons is supposed to be travelling along the 
x-axis with known momentum p. It is desired to observe an 
electron and to measure its position; for this purpose it is 
imagined that a microscope will be used, and since the utmost 
resolving power is required a short wavelength should be chosen. 
The position can then be determined to an accuracy given by 

Ax = A//a, 

where a is the aperture, A the wavelength, and / the distance 
from the electron to the lens. 



THE UNCERTAINTY PRINCIPLE 51 

Radiation cannot be scattered by an electron without dis- 
turbing the electron; radiation is scattered by free electrons 
according to the rules of the Compton effect, according to 
which the momentum lost by the light quantum when scattered 

is transferred to the electron. Thus 

i i 

if a quantum having frequency v, and 
hence momentum hv/c, is scattered 
through an angle 6, momentum equal to 

cos#)/c 




is transferred to the electron. Thus ^ 

we cannot observe the electron without * 

disturbing it. Moreover, we disturb it ** 

by an unknown amount, since, owing Fj rj 

to the finite aperture of the lens, 6 is 

not known exactly. In Fig. 13, may lie between ABC and 

ABC'. There is thus an uncertainty a/f in and hence, since 

0-90, of 7 ., 

/iva/cf 

in the momentum transferred to the electron. Since A = c/v 

this may be written A 7 . 

J &p = ha/jA, 

where Ap is the uncertainty in the momentum of the electron 
after the measurement has been made. We see that 

ApArr = h y 
as we expect. 



CHAPTER IV 
STATIONARY STATES 

1. THE OLD QUANTUM THEORY 

It can now be regarded as an experimental fact that the total 
internal energy of an atom or molecule is quantised. By the 
internal energy we mean the total energy of the electrons and 
nuclei moving about their centre of gravity; the kinetic energy 
of the translational motion of the atom or molecule as a whole 
can of course have any positive value. The internal energy, 
however, cannot have any arbitrary value, but only one of a 
series of discrete values, of which one is 

|- the lowest. This is what is meant by the 

statement that the energy is quantised. 

The simplest and most important appli- 
cation of this principle is to the energy 
of the electrons in an isolated atom, and 
thus in an atom of a monatomic gas or 
vapour. It is usual to measure this quan- 
tity with the convention that the total 
Fis?. 14. Ener<*v levels i -i ^ 

of an atom. ener gy 1S zero when one electron is re- 

moved far from the atom and is at rest. 
With this convention, the quantised energy values of an atom 
are negative. A typical scheme is shown in Fig. 1 4. The distance 
of each horizontal line from the zero represents the energy of 
the atom in one of the quantised states. The lowest state of 
the atom is known as the normal or ground state, the higher 
states as excited states. The energy 7 required to remove an 
electron to a slate at rest at infinity from an atom in the 
normal state is known as the ionisation potential. 

No detailed review of the experimental evidence for the 
existence of quantised states in atoms will be given here, but 
we may mention the following: 



THE OLD QUANTUM THEORY 53 

(a) The existence of a definite energy required to excite an 
atom, and the fact that it is large compared with the thermal 
energies")" of molecules in a gas, follow from the observation 
that the specific heat per gramme atom of a monatomic gas is 
fj?, which can all be accounted for by translational motion. 
Collisions between gas atoms therefore do not change the 
internal energy. 

(6) Many experiments J have been carried out which show 
that electrons after hitting an atom are deflected either without 
loss of energy, or with loss equal to one of the excitation 
potentials W l W QJ W 2 W of Fig. 14 or else greater than the 
ionisation potential /. 

(c) Detailed information about the energy levels is derived 
from spectroscopic evidence, coupled with the hypothesis that 
radiation of frequency v is emitted and absorbed in quanta 
according to the equation 



assuming this hypothesis, the existence of line spectra proves 
the existence of stationary states. 

It should be emphasised that only the isolated atom in a gas 
or vapour has a system of energy levels of the type shown in 
Fig. 14. The electronic system of an isolated molecule has a 
similar system, but in addition the vibrational motion of the 
nuclei about their mean positions and the rotation of the 
molecule as a whole introduce additional series of levels, much 
closer together. The energy levels of electrons in solids are not 
quantised (cf. Chap. V, 7). 

The hypothesis of the existence of stationary states was 
introduced into physics by Niels Bohr in 1913. At the same 
time he introduced another hypothesis in order to be able to 
calculate the values of the quantised energies for the case of a 

f The thermal energy \ kT of an atom at room temperature is 0037 of an 
electron volt (k T~ 1/40 eV.); the excitation potentials are of order 8-20 eV. 

J Cf., for instance, E. G. Dytnond and E. E. Watson, Proc. Roy. Soc. A, 
cxxn, 571, 1029. 

N. Bohr, Phil. Mag. xxvi, 1, 47tt, and 857, 1018. 



54 STATIONARY STATES 

single particle moving round a centre of force, or a pair of 
particles moving round their centre of gravity. This hypo- 
thesis is as follows: the orbits are as predicted by Newtonian 
mechanics, but only those orbits are found in nature for which 
the total angular momentum is a multiple of h/2Tr(= H). This 
hypothesis, though extremely valuable at the time, has now 
been abandoned in favour of the description given by wave 
mechanics. 

With the aid of this hypothesis one can show:f 
(a) That the energy of an electron moving round a nucleus 
of charge Ze and of infinite mass is 

_ ra 
~~~ 



n 



where n is an integer. This formula is confirmed by wave 
mechanics. Tt is in agreement with experiment, not only for 
the spectra of atomic hydrogen (Z = 1) and ionised helium 
(Z = 2), but for the X-ray levels of heavy atoms. For these it 
is a fair approximation to treat each K electron as moving in 
a field of a point charge (Z a}e, with a between and 1. 

(6) That if one takes into account the motion of the nucleus 
(mass M ) about the centre of gravity, m in the above equation 
must be replaced by ra*, where 



This small correction can be verified by comparing the values 
of the Rydberg constant obtained from hydrogen and ionised 
helium (cf. Chap. V, 2-1). 

(c) That a diatomic molecule rotating about its centre of 
gravity has quantised energy levels obtained as follows. We 
may treat it as a rigid body of moment of inertia 7, given by 



2a is here the distance between the nuclei and M the mass 
of each nucleus, the electronic mass being neglected. If the 

I Cf. the original papers by Niels Bohr, or any text-book on atomic physics. 



STATIONARY STATES IN WAVE MECHANICS 55 

angular velocity about the centre of gravity is o>, the angular 
momentum is Iw t so that Bohr's hypothesis gives us 

/co = Ifi, 
where / is an integer. The kinetic energy W is thus given by 



. (i) 

The treatment by the methods of wave mechanics replaces I' 2 
by /(Z-h 1), as shown in Chapter V, 2-2. 

It will be noticed that the interval between energy levels is 
less by a factor of order m/M (1/1860 for hydrogen) than for 
the line spectra of free atoms. For if we take for a the radius 
of the first Bohr orbit of hydrogen (S 2 /we 2 ) we find for the 
interval A W between the ground state (/ = 0) and the first 
first excited state (/ = 1) 

\W = ^- = m ^ 
27 2J/2S 2 " 

The second factor is the ionisation energy of hydrogen (13-60 
eV.). 

A W is thus less by a considerable factor than kT at room 
temperature (0-025 eV.). Therefore diatomic molecules in a 
gas in thermal equilibrium at room temperature will rotate, as 
is shown by the observed value |jR of the specific heat. At very 
low temperatures the specific heat of H 2 does in fact drop 
towards the value |Jf?, since when kT < W the collisions are not 
in general energetic enough to excite rotation. f Or, expressed 
in other terms, the exponential factor exp ( WjkT), which 
determines the number of rotating molecules, becomes small. 

2. TREATMENT OF STATIONARY STATES IN WAVE MECHANICS 

According to the principles of wave mechanics, any electron 
or other particle shut up in an enclosed space will have a 

t The experimental evidence is reviewed, for instance, by R. H. Fowler, 
Statistical Mechanics, 2nd ed., Cambridge, 1936, p. 83. 



56 STATIONARY STATES 

quantised series of energy values. Thus an electron bound in 
an atom, or an atom vibrating as a whole about a fixed position 
in a solid, will have quantised energy values, while a freely 
moving atom or electron will not. 

The reason for this can be seen most simply by considering 
an idealised case, that of the motion of a particle shut up in a 
box with perfectly reflecting sides. We need to consider motion 
in one dimension only; we thus consider that the particle 
moves along the x-axis from x to x = a, and that at these 
two extremities, where the walls of the box are, the wave 
function vanishes. The Schrodinger equation for the particle is 

2mW 



of which the solutions are 

sin kx, cos kx, 

wnere tc == ^^ 

The solution which vanishes at x = is sin kx, and this vanishes 
at x = a only if ka = UTT, and thus if 



_ _ 

" ~ " 2 ' * ' 



Solutions of the Schrodinger equation satisfying the required 
boundary conditions exist therefore only if W has the series of 
quantised values given by (2). 

It will be remembered that, according to wave mechanics, 
our knowledge of the position of a particle in a given state is 
given by a wave function ?/r, such that | iff \ 2 dx is the probability 
that the particle will be found between x and x + dx. In our 
case 

n . HTTX 
w ~ C sin , 
a 



STATIONARY STATES IN WAVE MECHANICS 



57 



where C is a constant. Such a solution can only be found if 
W has one of the values (2). We deduce that the energy can 
only have these values. 

The value of C should be chosen so that the function is 
normalised, i.e. so that 



fa 

^dx = 1. 
Jo 



As regards orders of magnitude, we note that if a is of the 
dimensions of an atom (3x 10~ 8 cm.), and m the mass of an 
electron, the quantity h 2 /Sma 2 is of the order of the ionisation 
potential of an atom (actually 4*1 eV.). If a is the diameter of 
a nucleus (say 5 x 10~ 13 cm.), and m the mass of a nucleon, we 
obtain c. 10 7 eV., of the order of the energies concerned in 
nuclear reactions. 

Suppose that we now \ f energy 3 
consider an electron shut 
up in a box bounded, not 
by perfectly reflecting sides, 
but by an electrostatic field 
which pushes the electron 
back when it tries to get 
out. The potential energy 
of an electron in this field 
is shown in Fig. 15. Let 
the electron have an arbitrary energy W. The wave function 
in the neighbourhood of the points A and B, where the 
'classical' electron would try to get out, is shown also in Fig. 
15; in the regions where the classical electron cannot go, $ will 
decrease exponentially, while within the box it will oscillate. 
If we start to draw the wave function from either end taking 
the solutions that decay exponentially outside the box instead 
of increasing exponentially, the two solutions will not in general 
join up in the middle; only for a discrete series cf energy values 
W n will they do so, and these will be the quantised values that 
the energy of the electron must have. 




58 



STATIONARY STATES 



An electron in a hydrogen atom is held in a box very much 
of this type. The potential energy of such an electron plotted 
along a line passing through the nucleus is shown in the upper 

part of Fig. 16: an electron with the 
energy represented by the horizontal 
line AB can move freely between 
the points A, B, where it will suffer 
total reflection. The wave function 
will be as shown in the lower half of 
the diagram; it will oscillate in the 
region between A and B and die 
away exponentially outside. Clearly, 
only for a series of energies W n will 
such a solution of the Schrodinger 
equation be obtainable. 




Fig. 16. Potential energy 
of electron and wave function 
for hydrogen atom. 



Exercise 

Write down an equation whose roots are the quantised 
energies of a particle moving along the x-axis in the field de- 
rived from the potential 

V(x) = |*|>cx, 
= - U | X | < a. 

Show that there is always at least one bound state for a 
particle in this field, but only one if 

(3) 



3. THE SIMPLE HARMONIC OSCILLATOR 

One of the simplest and most important examples in the 
theory of stationary states is that of the linear oscillator. 
A particle of mass M is held to a fixed point P by a restoring 
force px where x is the displacement. f According to 

f The most important applications are the vibrations of an atom in a solid, 
where x is the displacement, and the vibrations of a diatomic molecule, where 
? is the change in the internuclear distance. 



SIMPLE HAKMON1C OSCILLATOR 



59 



Newtonian mechanics, it will vibrate about P with arbitrary 
amplitude and energy, and frequency v given by 



= -1 IP 

27TV Jf" 



Our problem in quantum mechanics is to find the allowed 
quantised values of the energy W* and the appropriate wave 
functions. 




Fig. 17. Potential energy of a particle executing simple 
harmonic motion. 

The potential energy of the particle is \px*. This is plotted 
in Fig. 17; according to classical mechanics the particle will 
move backwards and forwards between A and B. Since the 
velocity of the particle is <j{2(W - V)m}, the chance that it will 
be found between x and x + dx is proportional to (W F)~*. 

The Schrodinger equation is 



It is interesting to obtain an approximate solution by means 
of the W.K.B. method. This is (for x between A and B) 



(5) 



60 STATIONARY STATES 

The true solution will decrease exponentially outside AB. As an 
approximation, however, we may demand that the wave func- 
tion shall vanish at the two extremities A, B. The lower limit 
of the integral in (5) must then be taken at A, and the condition 
is thus satisfied if 



x l being the values of x for which the integrand vanishes. 
Our quantum condition, then, consists in fitting n half waves 
into the interval AB. 

The integral can be evaluated by setting 

ipx* = IF sin 2 0; 
we find, as may easity be seen, 

W = nhv, n>,\. 

The exact solution of the problem involves finding the values 
of W for which a solution of (4) exists which oscillates between 
A and B and decays exponentially outside. These values of W 
are given bv 

W = 



The normalised solution! for the first two states are 



where 



[Mp\-* 

*-(*>) * 



The reader will easily verify that the above solutions satisfy 
(4). The quantity (H 2 IMp)* gives a measure of the radial 
extension of the wave function in the ground state. 

The quantum number n(n = 1,2, ...) which labels each sta- 
tionary state has in wave mechanics a simple meaning; n \ 

f For details of the solution cf., for example, Mott and Sneddon, p. 51. 



QUANTISATION IN THRIVE DIMENSIONS ()1 

is equal to the number of zeros in the imve function, not counting 
the zeros at the two extremities. 

It is worth emphasising here that, if we are dealing with an 
atom vibrating in a solid or molecule, the radial extension of 
the wave function in the ground state is small compared with 
the distance between atoms. For if this latter distance is a, we 
may expect the interatomic forces to be of order e 2 /' 2 , and thus 

p ~ e 2 /a?. 

The radial extension of the wave function, as we have seen, is 
*: substituting for^> this gives 



But H 2 /me 2 is the radius of the hydrogen atom, and thus of the 
same order as a. Thus the radial extension is of order 



where m is the mass of an electron, M of an atom. 

4. QUANTISATION IN THREE DIMENSIONS 

Quantisation in three dimensions differs from that in one in 
two respects: each stationary state is specified by three quantum 
numbers instead of one; and levels may be degenerate, that is 
to say two different states with different wave functions may 
have the same value of the energy. 

These two points are both illustrated by the case of a 
particle moving in a box with perfectly reflecting sides. Let 
the interior of the box be defined by 

0<x<a, 



< z < c, 

so that the box has rectangular sides with edges of length 
a, fe, r . The Schrodinger equation is 



62 STATIONARY STATES 

which has solutions satisfying the boundary conditions 



if, and only if, the energy W has one of the values 



2m la 2 ft 2 c 2 



(6) 



The quantum numbers n l9 n^ n%, with unity subtracted in each 
case, are equal to the number of nodal planes in the wave 

function parallel to the planes x 0, 
y = 0, and z respectively. The 
case illustrated in Fig. 18 shows the 
state (4, 3, 2). 

That degenerate states may occur 
is clear from formula (6). For in- 
stance, if a = b, the wave functions 
with quantum numbers (n v n 2 ,n 3 ) 
and (n 2 ,n l9 n 3 ) are solutions of the 
wave equation corresponding to the 

same value of the energy. Tt will be noticed that, when two 
degenerate states exist with wave functions ^ i/r 2 > then a linear 
combination such as 

(7) 



Fig. 18. Nodal planes of 
wave funetion of particle in 
a box. 



is also a solution of the wave equation. 

Exercise 

If a = b y sketch the nodal surfaces of the* wave function (7) 
for tij = 1 , w 2 = 2 and various values of the constants A , B. 

5. QUANTISATION WITH SPHERICAL SYMMETRY 
AND THK HYDROGEN ATOM 

This is an important problem, as it includes the treatment 
of the hydrogen and other atoms. We have to find the station- 
ary states of a particle moving in three dimensions which is 
under the attraction of a centre of force. The potential energy 



THE HYDROGEN ATOM 63 

of the particle is then a function only of its distance r from the 
centre of force. We denote it by V(r). In the particular case 
of an electron moving in the field of a positively charged 
nucleus of charge Ze, 

V (r) = =2*. (8) 

The Schrodinger equation is then 



-. 

The solutions of this equation in spherical harmonics can be 
found in numerous text-books and will not be given here. The 
salient points are given in the remainder of this section. 

The solutions may be divided into: 

(a) Solutions having spherical symmetry, such that 



The corresponding states of the atom are known as s states,f the 
quantum number /, giving the number of nodal surfaces passing 
through the origin, being zero. IU will later be identified with 
the angular momentum of the state. 
(6) Solutions having the forms 



These are known as p states; they have one nodal plane passing 
through the origin; thus by definition I = 1. A p state is triply 
degenerate, the three independent wave functions shown above 
all having the same energy. 
(c) Solutions having the forms 







f The symbols used in speetroscopy, ,/>, d, referred originally to lines, 
not states, and meant sharp, principal, diffuse. 



64 



STATIONARY STATES 



These are known as d states; they have two nodal planes 
passing through the origin, so that 1 = 2. The degeneracy is 
fivefold, not sixfold as would at first appear, because the three 
last wave functions are not independent, their sum being zero. 

(d) States of higher quantum number /; these can con- 
veniently be expressed only in terms of spherical harmonics 
=Pjii(cos 0) e iu *f(r) 

The degree of degeneracy is 21 -f 1 - 





2s 



/>!. 





3d 



Fig. 19. Nodal surfaces and amplitude of the wave functions of the 
hydrogen atom. 

The function /(r) will itself have a number of zeros; each of 
these determines a spherical nodal surface in the wave function. 

The principal quantum number, n, is defined so that n 1 is 
equal to the total number of nodal surfaces, planar and 
spherical. Thus n / 1 is equal to the number of zeros in f(r). 

If V(r) is given by (8) and thus is the potential energy in a 
Coulomb field, the energy of an electron depends only on n 
(apart from relativistic corrections mentioned in Chapter VII), 
and is given by we*Z 2 1 



THE HYDROGEN ATOM 



65 



This is not the case for any other field, the energy depending 
on I also. 

In Fig. 19 we show the wave functions of the hydrogen atom 
for a number of states. Above we show the intersection of the 
nodal surfaces with the plane passing through the nucleus. 

Turning now to the Schrodinger equation, it will easily be 
seen by direct substitution that for s states, where i/r =/(r), 
the equation reduces to 

d 2 f 2df 2w , T7V ^ 
4/1 J ' -^(W-V)f^O. 



The student is recommended to verify this. In the general 
case it becomes 



J 9 ' J 

dr* r dr 

The substitution 
reduces (9) to 



This is of the same form 
as the equation for the 
motion of an electron in 
one dimension. The solu- 
tion </, however, must van- 
ish at the origin, in order- 
that / may remain finite 
there. The type of solution 
is shown in Fig. 20. The 
quantity F(r) defined by 



,*- K) - 

* 



-0. 



(9) 



(10) 




*\r) = -jp( w ~ v )- ^2 F^ 20. 

is positive between two values r 1 and r 2 ; within this region the 
wave function oscillates; outside it decays exponentially. 



66 STATIONARY STATES 

For the case of the Coulomb field, equation (9) may be 
solved in series. The solution may be found in any text-book; 
we quote here the first three wave functions 

1* (n= 1,!0) /(r) = qe-^, 

2s (n = 2, / = 0) /(r) = c 2 (2--)e-^ 2 , 

a 

22) (n = 2, I = 1) /(/) = c 3 re- r ' 2a . 
Here a is the Bohr radius, given by 

a = 



and the constants C 15 c 2 ,c 3 are normalising factors. The student 
is recommended to verify by substitution that these values of 
f(r) are solutions of equation (9) for the appropriate values of W. 

Exercises 

(1) Consider the motion of an electron in the field defined by 

F(r) = r>cx, 



Show that there are no stationary states at all if 



(2) For the case / = and V(r) = -e 2 /r, the solution of (10), 
vanishing at the origin, found approximately by the W.K.B. 
method, is 



Remembering that W is negative, obtain approximate values 
of the quantised energy values by fitting n half- waves into the 
region where W V is positive, i.e. by setting 



where r is the value of r for which the integrand vanishes.f 

t The method in fact gives the exact value of W. This would not l>e the 
case for any other form of V(r). 



INTERPRETATION OF WAVE FUNCTIONS 67 

The integral can be evaluated by setting 



(3) Show that if F(r)-> const. /r at large values of r, there 
will exist an infinite series of stationary states leading up to a 
series limit; while if V(r) tends to zero faster than r"" 1 , the 
number of bound states is finite. 

(4) Show that, for an attractive field of the type 

the equation giving the energy values is indeterminate. 

0. INTERPRETATION OF THE WAVE FUNCTIONS 

If a hydrogen atom is, for example, in the ground state, it is 
described by a wave function ^ given by 

i/j(r) = Ce~ rta , a * W/me* = 0-54^4. 

The interpretation of this function is, as usual, that \^(r) \*dr 
is the probability that an electron will be found in the volume 
element dr at a distance r from the nucleus. This is all the 
information provided by wave mechanics about the position 
of the electron. 

It is usual to choose the constant C so that 

(r)| 2 dr=l; (11) 

the wave function will then describe a state of affairs where it 
is known that an electron is in the atom. The wave function is 
then said to be normalised. It will easily be verified that 

/~t _ _ A ._ } 

w - 7T W 

In some phenomena (e.g. scattering problems, cf. Chap. II, 
7) the atom behaves like a nucleus surrounded by a cloud of 
negative charge of density 



68 STATIONARY STATES 

where */> is the normalised wave function. The electrostatic 
potential of the field due to a nucleus and this charge is, as 
may easily be verified, 



,(VW 

\r a! 



A second electron in the neighbourhood of a hydrogen atom 
may thus be treated as though it moved in a field of potential 
energy V(r) such that 



In atoms containing more than one electron it is a fair 
approximation to consider each electron as having its own wave 
function ifj(r). Thus, for instance, in the helium atom each 
electron may be regarded as having a wave function 



where a is a parameter. One way of finding the best value of 
the parameter a is the variational method given in the next 
section. 

Another method of finding the wave functions ifj(r) is that 
due to Hartree, the method of the 'self-consistent field'. 
Hartree does not use an analytic form for 0(r); for the helium 
atom he proceeds as follows. He supposes that each electron 
moves in a field of potential energy V(r) produced by the nucleus 
(charge Z = 2) and by the other electron, treated as a distribu- 
tion of charge of density e \ ifj(r) | 2 . V(r) will then be given by 
Poisson's equation 



dr 2 r dr 
and the boundary conditions 

V(r) ~ - 2e 2 /r r small, 
-- e 2 jr r large. 



VARIATIONAL METHOD 69 

If we know i/r, then, we can calculate F(r); and if we know F(r) 
we can calculate if/ from Schrodinger's equation. Hartree, by a 
process of successive approximations, calculates ifj so that it is 
self-consistent; in other words, if V is deduced from i/r and then 
iff calculated, the original value is obtained. f 

7. VARIATIONAL METHOD OF OBTAINING APPROXIMATE 
WAVE FUNCTIONS 

Let us write the Schrodinger equation for a single particle 
in a field with potential energy V(r) in the form (cf. p. 46) 



ft 2 

where # = ~ '- V 2 +F. 

2m 

Multiplying by 0* and integrating over all space we see that 

W = fy*H*l*dT. (13) 

The total energy of the electron is thus the sum of two parts: 

(a) |j/r*Fi/fC?T. This clearly represents the potential energy of 
the electron in the field of potential energy V(r). 

Jj2 / 

(b) r ;-- \\fi*V 2 tf/dr. This (positive) term represents the kin- 

2w J 

etic energy. The reasons why this is so lie somewhat outside 
the scope of this book. 

It is shown in the more advanced books on wave mechanics 
that if ifj(x,y,z) is any normalised function of #,2/,z, i.e. such 
that (11) is satisfied, then the true wave function is that for 
which the integral (13) is a minimum. This gives a very useful 
method of obtaining approximate wave functions; one can take 
a given analytic expression containing one or more parameters, 
work out the integral (13) and choose the parameters so as to 
minimise the energy. 

f Hartree's method is discussed further in Chap. V; a review is given by 
D. R. Hartree, Rep. Prog. Phys. xi, 113, 1940-7. 



70 STATIONARY STATES 

Exercises 

(1) For the hydrogen atom, take of the form Ce~* r normal- 
ised to unity and work out the integral (13). Show that it has 
the minimum value when A = me 2 /H 2 . 

(2) A hydrogen atom is placed in a strong electric field E, 
so that the potential energy of its electron is 

e 2 

--- 1- Eex. 
r 

By assuming a wave function of the form 



obtain an approximate expression for the energy due to the 
field. 

(3) Describing both electrons in the helium atom by a wave 
function of the form (12) with an unknown value of a, 
calculate the first ionisation energy. This is done as follows. 
First calculate the kinetic energy and potential energy in the 
field of the nuclei. Their sum is 



the 2 arising because there are two electrons. Then calculate 
the interaction energy of the electrons; this, by the arguments 
of the last section, is 



r i /i i\ 

U* -e 2 (~ + e~ 
J I \r a) 



The sum of these is the total (negative) energy of the atom, 
and a must be chosen so that it takes a minimum value. To 
obtain the first ionisation energy, we must subtract it from the 
second ionisation energy, viz. 4 x we 4 /27& 2 . 

8. ORTHOGONAL PROPERTY OF THK WAVE FUNCTIONS 

This property is of considerable importance for subsequent 
developments. The term orthogonal means that, if lf i// 2 are 



ORTHOGONAL PROPERTY 71 

solutions of the Schrodinger equation corresponding to two 
different energy values, then 



the integral being over all space. 

Exercise 

The student is recommended to verify the orthogonal rela- 
tion for the functions shown on pp. 60, 66. It is, for instance, 
obvious from symmetry that s and p functions are orthogonal 
to each other. 

We shall prove the orthogonal relation for the case of one 
dimension only, that is to say for a particle moving along a 
straight line. The two wave functions then satisfy the equations 



If we multiply the first equation by $$ and the second by 
and subtract the second from the first, we obtain 



The first two terms may be written 



Thus on integrating over all space 

2m 



The term on the left vanishes, because, since we are dealing 
with bound electrons, the wave functions tend to zero far from 
the region in which they are bound. Also we have postulated 



72 STATIONARY STATES 

that Wj and W 2 should be unequal. It follows that 



i: 



We have taken the complex conjugate of 2 rather than the 
function itself. For a particle moving in an electrostatic field 
the wave functions are real, and this makes no difference. In 
the presence of a magnetic field, however, terms of the type 
Aidy/dx occur in the wave equation (Chap. I, 7), so the wave 
functions are complex. The reader will easily verify from Chap- 
ter II (13) that if A is a constant the relation (14) is verified 
in this case. 

An important consequence of the orthogonal relation is that 
any arbitrary function /(#, y, z) can be expanded in a series of 
wave functions i// 7l , provided that the function / satisfies the 
same boundary conditions as the functions ^ 7J . Thus suppose, 
for instance, that the functions $ n (x,y,z) are the series of 
wave functions for the hydrogen atom, and that f(x, y, z) is any 
function which tends to zero at infinity. Then we may write 

n (x,y y z), (15) 



and find the coefficients A n by multiplying both sides by $* 
and integrating over all space. We find, using the orthogonal 
relation, that 



For a particle bound in a box, a simple harmonic oscillator, 
etc., all the states are quantised. For a hydrogen atom, on the 
other hand, the summation should include an integration over 
the unquantised states, in which the electron is free from the 
atom (cf. Chap. VI, 3). 

It will be shown in Chapter VI that if an electron is des- 
cribed by a wave function of the type (15), the coefficients A n 
may be interpreted as follows: we do not know in which 
stationary state the electron is; the chance that it is in the 

is . 



PERTURBATION THEORY 73 

9. PERTURBATION THEORY 

It is very useful to be able to calculate the change in the 
energy of an electron in an atom due to a small perturbation, 
for example, an electric or magnetic field. This may be done as 
follows : 

We write the wave equation 

(H-W)t = 0, (16) 

and calculate the change w in W due to a small additional 
term v(x,y,z) added to the potential energy. The wave equa- 
tion may now be written 

{H + v-(W + w)}(t+f) = 0, (17) 

where / is the change in i/j. Making use of (16) and neglecting 
all terms of the second order such as vf, wf, this gives 

(H-W)f+(v-w)$ = 0. 
Now we may expand/ in a series of the type (15) 



where the ifj n are the solutions of (16) and W n the corresponding 
energy values; we use also the suffix zero to denote the state 
the electron is in. 
We then find 



If we multiply both sides by $$ and integrate, we find 

w= 0*t^ rfT, (19) 



a formula which is essentially the same as (13), and gives us to 
the first order the change w in the energy. If we multiply both 
sides by 0j* and integrate, we obtain 



(20) 



rr w ~~ rr O 

which, with (18), gives us the perturbed wave function. 



74 STATIONARY' STATES 

Exercises 

(1) Let us suppose that the potential energy of an electron 
in the field of a nucleus is, owing to the finite radius r of the 
nucleus, 



r<r . 

Calculate the change in the energy of the Is state due to this 
correction. 

(2) A particle is bound to its mean position by a force such 
that its potential energy is %px* + qx*. Treating the second 
term as a perturbation, find the energies of the first two 
stationary states. 

Two of the most important applications of perturbation 
theory are to an electron in an electric and in a magnetic field; 
these will be treated in the next three sections. 

10. THE POLAR1SABILITY OF AN ATOM 

In this section we shall calculate the polarisability of an 
atom. The calculation will be specifically for a hydrogen atom, 
but can be extended to any atom if we describe each electron 
by its own wave function (p. 102). 

We give first a classical calculation. Suppose that an electron 
is bound to a mean position by a force px when the displace- 
ment is x. It can then vibrate with a frequency V Q given by 



In the presence of a field E, the displacement x is given by 

px = eE. (21) 

The dipole moment is thus 

ex = e*E/p = e*E/47T 2 mvQ. 
The polarisability a, defined as ex/E, is thus given by 

'* (22) 



POLARISIBILITY OF AN ATOM 75 

If, instead of a static field, E is replaced by a field J cos2m> 
oscillating with frequency v, (21) becomes 

mx + px = e E Q cos "l^vt ; 
on integrating we find 

ex = e 2 E Q cos 27n//47r 2 ra(i/ 2 - y 2 ), 
so that the polarisability is now given by 

e 2 

* = 4^^vf)- (23) 

This calculation is artificial in that it makes use of the con- 
cept of an elastically bound electron, capable of vibrating with 
a definite frequency. We shall now give the corresponding 
wave-mechanical calculation. 

The electron in .the atom is acted on by a field E in, for 
instance, the z direction. The addition term in the potential 
energy of the electron, due to this field, isf 

v(z) = Eez. 

In the presence of the field, then, the wave function of the 
electron is, instead of the original ^ (r), by (18) and (20) 



-V *(* .V> -) (24) 

n H'w "~ n O 

where 2 n0 is defined by 



and dr denotes the element of volume dxdydz. The charge 
density p(x,y,z) in the atom is obtained by taking the square 
of the modulus of (24) and multiplying by e, so that 



p(x,y,z) = _*|^(r)! + etfS=25= W,>l> + tM (25) 

n "n ~~ ^0 

Terms of order E 2 are neglected, since the polarisability is 
always defined for values of E small compared with the fields 
within the atom. 

t Note that e is the numerical value of the electronic charge, - e the charge 
on the electron. 



76 STATIONARY STATES 

Now the dipole moment of the atom is by definition 

Jp(x,y 9 z)zdr 9 (26) 

so that the polarisability a is Ipzdr/E. The function | ift (r) | 2 is 

spherically symmetrical; thus, substituting (25) in (26), the 
first term gives a vanishing contribution, and we are left with 



which gives us the desired quantum mechanical formula for the 
polarisability of an atom. 

This formula (27) may be compared with the classical 
formula (22). The frequencies v nQ of the absorption lines of 
the atom are given by 

Thus (27) may be written 



, 

xvn*>rrk 

wnere 






It may also be shown, though the calculation will not be 
given here, that for a field vibrating with frequency v, the 
formula for the polarisability which generalises (23) is 

-/ n -S- (28) 



The quantity f n0 is called the 'oscillator strength' of the 
transition. It may easily be provedf that, for atoms containing 
a single electron, 



That this relation is satisfied follows also from (28), since for 
high values of the frequency v the polarisability must tend to 

t Of., (or example, Mott and Sneddon, p. 169. 



POLARISABTLIT Y OF AN ATOM 77 

the classical value (23), the forces binding the electron to the 
atom being then unimportant. 

It will be shown in Chapter VI that the intensities of absorp- 
tion and emission lines are proportional to the oscillator 
strengths. 

It will be seen at once by symmetry that for the hydrogen 
atom in tho ground state all oscillator strengths vanish except 
those for which n refers to a p state. The values of the non- 
vanishing oscillator strengths have been calculated. f Some 
values are, for transitions from the initial state Is: 

Final state f n0 , 

n = 2 0-4161, 

n = 3 0-0791, 

Asymptotic formula for large n l-6n~ 3 , 
L/ w for all discrete values of n 0-5041, 
S/ H for continuous spectrum 0-4359. 

The dielectric constant *, equal to the square of the refrac- 
tive index, /n 2 , can be deduced from the value of a by the 

formula 2 i . A \r 

K n* = I 4- 47riVa, 

where N is the number of atoms per unit volume. It will be 
seen that, if v^>^ n() , this formula tends to 



the usual formula for a medium containing ^V free electrons 
per unit volume. 

It is also of interest to calculate the amplitude of the radia- 
tion scattered by a single atom. In the presence of an oscil- 
lating field E, the dipole induced in the atom is E. The 
electric vector of the scattered radiation, at a point P at a 
distance r from the atom in a direction making an angle $ with 

E ' is 



t Cf., for instance, H. Bethe, Handb. Phys. xxiv, pt. 1, 443, 1933. 



78 STATIONARY STATES 

Inserting formula (28) for a we find for the electric vector of 
the scattered radiation 

_ y Sin <Z> 2^ ^ n 

rmt* Y n 2 -v 2 

It will be noticed that, if the incident radiation is on the 
long wavelength side of all absorption lines, the scattering 
increases with decreasing wavelength. 

If v5>v H Q for all absorption lines of appreciable oscillator 
strength, the formula becomes 

E e 2 

-- -sin^, (30) 

r me 2 

the scattering formula for a free electron. Such a formula is 
obviously only valid if the wavelength of the radiation remains 
great compared with the size of the atom. For the case when 
this is not so, compare Chapter V, 46. 

Exercise 

Work out the oscillator strength for the transition 2p to 1 
of the hydrogen atom, using the wave functions given on 
p. 66. 

11. THE STARK EFFECT 
> 

The threefold degenerate p states are split by an electric 
field. This may be seen as follows. The energy of the atom due 
to the electric field is %aE 2 . If the atom is in a p state, the 
wave functions are of the form xf(r) y yf(r), zf(r). Thus, if the 
field E is along the z-axis, the oscillator strength of the transi- 
tion p~+ls, for instance, vanishes except for the last of these 
wave functions. Thus for a field in the z-direction, the polarisa- 
bility of the state zf(r) differs from that for the other two. 
The three p states therefore split into one non-degenerate and 
one doubly degenerate state. 

Consequently p~>s transitions appear as doublets. 



KFFECT OF A MAGNETIC FIELD 79 

12. EFFECT OF A MAGNETIC FIELD 

In this section we determine the effect of a magnetic field on 
an atom in a p state, obtain an expression for the Zeeman effect 
for an electron without spin, introduce the electron spin and 
describe the spin doublet in X-ray and optical spectra. 

Suppose a magnetic field H is applied along the z-axis. Then 
there are three ways in which we can treat it: (a) by classical 
theory, (b) by old quantum theory, (c) by wave mechanics. 

(a) By classical theory. We suppose (as on p. 74) that 
w the electron is held in position by an elastic force pr for 
a \displacement /*, so that it can vibrate with frequency 
V Q =4- (27r)~\/(/>/n?). In the presence of the magnetic field there 
are t'hree possible normal modes, with different frequencies. 
The electron can vibrate along the field, in which case the 
frequency is unaltered, or rotate in circular orbits in the plane 
perpendicular to the field. In the latter case, if w is the angular 
velocity v of the motion and r the radius, we have 

wo* 2 r = pr eHwr/c. 

The firsjt term represents the centrifugal force, the last the 
force act\ing on an electron moving with velocity o>r perpendicu- 
lar to a ^nagnetic field. Solving for to, and treating H as & 
small quantity, we find 

,. J*.. (31) 

2n 4-Trmc 

If emission or absorption lines were due to vibrating elec- 
trons, we should' expect that a magnetic field would split all 
lines into three components. Such a splitting is observed in a 
magnetic field (Zeaman effect), but only for singlet lines is it 
given by (31). The historical importance of the Zeeman effect 
is that the occurrence of e/m in a formula in agreement with 
experiment in certain cases gave the first experimental proof 
that electrons take part in the emission of light from atoms. 

(6) By old quantum theory. One can show that the magnetic 
moment of an atom ii which an electron rotates with angular 
momentum / is e//2wc. For a circular orbit the proof is simple. 



80 STATIONARY STATES 

The magnetic moment is the product of the area (rrr 2 ) and the 
current in electromagnetic units, which, if e is as usual in electro- 
static units, is equal to 

7TT 2 x eoj/2Trc = ea)r 2 /2c el/2mc. 

The magnetic moment of an atom with angular momentum in 
is thus , 

en i. 

2mc 

The quantity eh/2mc is known as the Bohr magneton and will 
be denoted by ju#. 

If one supposes that the component uU of the angular 
mentum along the magnetic field is also quantised and u 
integral values, it follows that the energy of the atom iija ' 
magnetic field is 



The level thus splits in 2/+ 1 states (cf. 5). 

Making use of the selection rule (Chap. VI, STM^flHP in 
optical transitions u will change only by or 1, WMrthat 
EI spectral line of frequency V Q will, in the presence of t itoagnetic 
field, split into three lines; the frequency of on of ihese is 
unaltered, while those of the others differ from i^-by v where 

H. 



The predictions of the old quantum theory for an electron 
without spin are thus the same as (31). 

(c) By wave mechanics. This again gives the /same result, and 
identifies the angular momentum Ifi of the last section with the 
quantum number / of p. 64. We shall, however^ confine ou; 
;o p states (I 1). ^ ;* 

We have expressed the three wave functions 
n the form 




Taking spherical polar coordinates tMs| can, however, be 

vritten 

sin 6 cos <f>f(r), sin 6 sin 6f(m/B cos Of(r]. 




THE ELECTRONIC SPIN 81 

<f> is here the azimuthal angle about the magnetic field. If, then, 
we want to represent rotation of the electron about this field, 
it is clear that the correct combinations of these wave functions 
to represent the three normal modes as in (a) above are 

0J = sin OeWffr), 02 = sin 0e-**f(r), 3 = cos 6f(r). (32) 

Now the perturbing term due to a magnetic field H along 
the z-axis is (cf. Chap. II, 7) 

eH 



rr 

U l7kt>~ 7 , CO = r - , 

d<f> 2mc 

and the changes in the energy of these three states due to the 
field are thus 

n= 1,2,3. 



/ 

\^U^ n 
v 



functions being normalised, this gives for the three 
states of (32) Hjigt 0> 



the same result as on the old theory. 

This treatment, moreover, identifies I with the angular 
momentum of the atom. 

13. THE ELECTRONIC SPIN 

The hypothesis that the electron possesses a mechanical 
moment (angular momentum) equal to one half quantum (|^), 
and a magnetic moment fi B (one Bohr magneton) has to be 
introduced into physics for the following reasons: 

(a) According to wave mechanics atoms in which one electron 
is outside a closed shell (e.g. Na, Ag) should in their normal 
states have the quantum number I equal to zero and should 
thus have no magnetic moment. The experiments of Gerlach 
and Stern on the splitting of beams of atoms by an inhomo- 
geneous magnetic field show, however, that for the silver atom, 
for example, the ground state splits into two in a magnetic 
field. Since the multiplicity of a state with angular momentum 
in is 2/-fl, one has to assume that the angular momentum of 
the atom is i^, and ascribe this to the electron itself. 



82 STATIONARY STATES 

(6) Spectroscopic evidence shows that states for which I > 
in atoms with one electron outside a closed shell are doublets. 
This can only be ascribed to the electron spin, which can have 
two orientations in the internal magnetic field which results 
from the orbital movement of the electron. 

(c) Measurements of the gyromagnetic effect of ferromagnetic 
materials enable a value to be obtained for the small change in 
angular momentum of a specimen that accompanies a change 
in magnetic moment. For iron and nickel the ratio of magnetic 
to mechanical moment is e/mc, not ef^mc as would be the case 
if the magnetism were due to the movement of the electrons. 
This proves the existence of elementary magnets (the electrons) 
for which the ratio is efmc. Since the mechanical moment is |ft, 
the magnetic moment is ek/2mc, or one Bohr magneton. 

In describing the spin, then, we need to introduce a variable 
a z which can only take two values, 1. a z jji B H is defined as 
the energy which the electron's spin will have, if a magnetic 
field H is set up along the z-axis. It amounts to the same 
thing to say that \Tta z is the component of the mechanical 
moment of the spin along the z-axis. The state of the electron 
will be described by a wave function x(^)*> the interpretation 
of this wave function is as usual that | x( a z) I 2 gives the proba- 
bility that, if a measurement were made to determine the 
energy of the spin in a field H along the z-axis, the result 
would be Hjji B <j z (a z = 1). There are two stationary states 
for x(z)'> t>he first, x a ( a s)> is defined by the equations 



and describes the state of the spin when the energy is known 
to be + nxH. The second, xp( a z)i 1S defined by 



and describes the state of the spin when the energy is known 
to be fi s H. 

In this book these wave functions will be used only in 
describing the two-electron problem (cf. Chap. V, 3). 



THE ELECTRONIC SPIN 83 

The complete description of an electron is by the product of 
the orbital wave function </r(r) and the spin wave function 



gives the probability that the electron is in the volume element 
dr at the point r, and at the same time the spin moment along 
the z-axis is \a$i (a z = I). The orbital function ^(r) will be 
little affected by the spin unless the electron is moving with 
velocity comparable with that of light, which for electrons 
bound in atoms is only the case for the inner X-ray levels. 
That the effect is small may be seen most simply as follows. 
If an electron is moving with velocity v it produces a magnetic 

field of order Tr . 9 

H = ev/cr 2 . 

The energy of the electron's magnetic moment eh/2mc in such 
a field is of order 



But, if r is the radius of the atom, h/mr is of the order v, where 
v is the velocity of an electron in the atom. Thus the energy 
term duo to the electron's spin is of order 



which is smaller by the factor v*/c 2 than the energy e 2 /r of the 
electron in the field of the rest of the atom. 

If we substitute v~e 2 /Ti, the term # 2 /c 2 is seen to be of order 
(e 2 /ftc) 2 . The quantity e 2 jHc is known as the fine structure 
constant, and is equal approximately to 1/137. 

For the calculation of the interaction between spin and 
orbital moment and for the evaluation, for instance, of the 
splitting of p states, one would naturally use the more exact 
theory of the electron due to Dirac, and reviewed briefly in 
Chapter VII. The analysis given here and the wave functions 
X(<T) are more convenient, however, for an elementary treat- 
ment of the two-body problem, as given in the next chapter. 



CHAPTER V 
THE MANY-BODY PROBLEM 

1. THK WAVE EQUATION FOR TWO PARTICLES 

The problem considered in the preceding chapters has been the 
motion of a single particle in a field of force. The state of a 
particle has been described by an 'orbital' wave function 
ifj(x, y, z) which depends on the spatial coordinates x, y, 2, and 
by a spin wave function x(z) depending on the component a a 
of the spin moment along the z-axis. In this way a discussion 
of the hydrogen atom has been given by treating the electron 
as moving .in the field of a fixed proton; and a discussion of 
more complicated atoms has been given by treating each elec- 
tron as moving in the field of the nucleus and the averaged field 
of all the other electrons, so that each electron is given its 
separate wave function. This method is, of course, an approxi- 
mation; in this chapter, then, we shall develop the theory 
appropriate to several interacting particles. 

Let us consider two particles of masses m l9 m 2 moving along 
a straight line, and having coordinates x l9 x 2 . Suppose also 
that the potential energy of the system when the two particles 
are at the points x l9 x 2 is V(x l9 x 2 ). Then according to classical 
mechanics the equations of motion are 



3V BV 

- , m 2 

CXi 

Now if we make the transformation 



m l x l =. - , m 2 x 2 = - 

CXi 



these equations transform into 



\VAVE EQUATION FOB TWO PARTICLES 85 

and are thus the same equations as those of a single particle of 
unit mass moving in two dimensions with coordinates g l9 2 . 
This suggests that in wave mechanics also the treatment of 
two particles each moving on a straight line should be the same 
as that of one particle moving on a surface. If this is so the 
two particles will be described by a wave function i//(x lt x 2 ) y of 
which the interpretation will be the following: | if/(x l9 x 2 ) \ 2 dx l dx 2 
is the probability that at any moment one particle will be 
found with its coordinate between x l and x 1 -h dx l and the other 
particle between x 2 and x 2 -hdx 2 . Also this wave function will 
satisfy the equation of a particle of unit mass moving in a 
plane, namely 

' 



or, transforming back to the coordinates x l9 x 29 



The treatment can be extended to the problem of two par- 
ticles moving in three dimensions. Their behaviour should be 
determined by a wave function ^(x^y^z^x^y^z^) of the co- 
ordinates of both particles. The interpretation of the wave 
function is that, if 

P = i VH^i, </i> *r, * 2 > y* z z) ^dr^dr^ (2) 

then P is the probability that one particle will be found in the 
volume element dr l at the point (x^y^z^ and the other in the 
volume element dr z at the point (x ZJ y 2 ,z 2 ). The wave function 
satisfies the equation 



which is the Schrodinger equation for a pair of particles. V is 
the potential energy of the particles, both in one another's 
field and in any external field. 



86 THE MANY-BODY PROBLEM 

The wave function for a pair of particles is thus a function 
of six coordinates. It will be realised that the 'wave' repre- 
sented by this function is not a wave in any medium with 
extension in space. 

It will be seen that if the two particles do not interact, and 
if one is in a state defined by a wave function ^ a (x l9 y ly 1 ), and 
the other in a state defined by a wave function $ b (x^y& z 2 ), the 
wave function for the pair of particles is the product 

t( x i> 2/i V> X 2> 2/2> *a) = 0(*i> Vi> z i) tb( x * 2/2* %)- ( 4 ) 

This is consistent with the interpretation of the wave functions, 
and can also be deduced from the wave equation, as the reader 
will easily verify. 

If the two particles are of the same type (two electrons or 
two protons), the form is more complicated than (4), (cf. 3). 

2. A PAIR OF PARTICLES IN ONE ANOTHER'S FIELD 

Under this heading we include 

(a) the hydrogen atom, when the motion of the nucleus is 
considered, 

(b) the rotation of diatomic molecules; in an elementary 
treatment we may suppose that the effect of the electrons is to 
introduce a force holding the two nuclei at a certain distance 
from each other. 

The potential energy function V in (3) is then a function 
V(r) of the distance r between the nuclei. The wave equation 
(3) can be separated by writingf 



(X, Y,Z) are then the coordinates of the centre of gravity of 
the two particles, (x, y, z) the coordinates of one particle relative 

f For a detailed treatment of this transformation, cf. A. Sommerfeld, 
Wave Mechanics, London, 1930, p. 27. 



PARTICLES IN ONE ANOTHER'S FIELD 87 

to axes through the other. The equation transforms into 

H* i 8* & d*\ 2 / P 8* d*\ 
^A^ + ^ + ^r^m^ 

where M Q = m l 4-m 2 , m* = m l m 2 /(m l + ra 2 ). 

Solutions of this equation may be obtained having the form 



h 2 
where / satisfies -^ V 2 /+ WJ = (6) 



and g satisfies ~ ^ V 2 g + (W 2 - V) g = (6) 

if Wi + W 2 = IF. Clearly/ is the wave function of a free particle 
of mass M, and represents the movement of the atom or mole- 
cule as a whole; g describes the probable length and orientation 
of the line joining the particles. Equation (6) is the same as 
that of a particle moving in a field in which its potential energy 
is F(r), except that w* replaces the mass m of the particle. 

2 1 . The hydrogen atom 

Equation (6) shows that the quantised values of the internal 
energy W 2 of an atom consisting of a nucleus of mass M and 
charge Ze and an electron are 

W = - 

where m* = m/(l+ m/M). e,h, and m are scarcely known accur- 
ately enough for the difference between m and m* to be 
observable directly in the spectrum of hydrogen; but the differ- 
ence in the Rydberg constants as deduced from the spectra of 
hydrogen (Z = 1) and ionised helium (Z = 2) enables the effect 
to be seen and a value of m/M obtained in good agreement 
with that from other sources. f 

f See, for instance, J. A. Crowther, Ions, Electrons and Ionising Radiations, 
7th e<i., Ixmdon, 1944 p. '274. 



88 THE MANY-BODY PROBLEM 

2-2. The diatomic molecule 

We have here at least three particles to consider: the two 
nuclei, and one or more electrons. In what follows we shall as 
usual denote the mass of each electron by m and the co- 
ordinates of all of them by the single symbol q. The masses of 
the two nuclei will be denoted by M l9 M 2 and their coordinates 

by 

^ = (^,7^) 

and R 2 = (X Z ,Y Z ,Z Z ). 

We write the distance between the nuclei as 



It was first shown by Born and Oppenheimerf that it is 
permissible, to a good approximation, to treat the molecule in 
the following way. First we solve (or imagine solved) the 
Schrodinger equation for the electrons moving in the field of 
the nuclei supposed at rest at the points R x , R 2 ; the solutions 
will be a series of wave functions ift n (R l9 R 2 ; q) with correspond- 
ing energy values W n (R). We then treat this energy W n (R) as 
though it were part of the potential energy of the two nuclei 
when distant R apart. In fact we take for this potential energy 



(7) 



thus adding to W ri (R) the Coulomb interaction of two nuclei 
with charges Z^, Z 2 e. 

A discussion of some methods of calculating V(R) will be 
given in 6; we obtain, for normal and excited states, curves 
such as shown in Fig. 21 ; for a stable molecule the ground state 
has a minimum, shown at the point P. The value R for which 
this occurs is the equilibrium value of the distance between 
the nuclei. 

t M. Born and J. R. Oppcnheimcr, Ann. Phys., Lpz. LXXXIV, 457, 1927. 



DIATOMIC MOLECULE 



89 



Following the method of Born and Oppenheimer, we then 
tfrite down the Schrodinger equation for a pair of nuclei acting 
)n each other with a force such that their potential energy is 
V(R). This is 



- V(R)}<fl = 



(8) 



V(R) 




Fig. 21. Energies l'(R) of a diatomic molecule as a function of 
the distance R between the nuclei; curve (1) is for the ground state, 
curves (2) and (3) for excited states. The vibrational wave functions 
for the first two excited states are shown below. 



Just as for the hydrogen atom the wave function can be 
separated into a factor / describing the motion of the centre 
of gravity, and a factor g(R) describing the behaviour of the 
line joining the nuclei, that is to say, the vector R. It is this 
second function (/(R) which is of interest; it describes rotation 



90 THE MANY-BODY PROBLEM 

of the molecule about its centre of gravity, and vibration about 
the mean position P. The equation satisfied by y is, writing M 
for J/(l/ 1 + 3/ 2 ), 

O f (9) 

which, as for the hydrogen atom, has solutions of the form 
(cf. Chap. IV, 5) 



g(R) = / 
where G(R) satisfies 

2M l(l+ 

- W - F ~ 



This solution describes a state of the molecule in which it is 
rotating with / quanta of angular momentum. 

Now the function V(R) has, we assume, a minimum for 
R = R . In the neighbourhood of this value of JR, then, we 
may write 

rw-i/j-i + ^W). (..) 

where a is a dimensionless constant, in general of order unity. 
U is the dissociation energy of the molecule, for most molecules 
in the range 2-6 eV. U is thus of the order of the excitation 
potential of an atom, and R () of the atomic radius. But M is 
several thousand times greater than the electronic mass. Thus 
the arguments of Chapter IV, 3, show that the radial extension 
of the wave function (?, for the first few states at any rate, is 
small compared with 7? . It is thus legitimate to use the 
approximation (11) for V(R), and also to replace R by R Q in 
the term 1(1+ \)/R* in (10). Equation (10) thus becomes 



= 0. 



THE FEANCK-CONDON PRINCIPLE 91 

Comparing this with (4) of Chapter IV, we see that x may be 
replaced by R R and p by 2f7a//?g: the quantised values of 
the energy W are given by 



where n in an integer and 



Various points may be noted about this formula. The term 
h 2 l(l+l)/2MRl represents the rotational energy of a molecule 
with moment of inertia M R% and I quanta of angular mo- 
mentum; it replaces the formula ft 2 / 2 /23/J?g [Chap. IV (1)] of 
the old quantum theory. It will be noticed that the interval 
between energy levels is smaller by a factor of order ni/M than 
the interval between energy levels of an atom. The last term 
in (12) gives the vibrational energy. The interval hv between 
vibrational levels is of order ^( M jm) larger than that between 
rotational levels. The energy levels of a molecule are thus 
crowded far more closely together than those of an atom and 
give rise to the so-called band spectra. 

2-3. The Franck-Cond,on principle 

This states that if a molecule is in a vibrational state such 
as that represented by the horizontal line AB in Fig. 21. and 
if through absorption of radiation it makes a transition to an 
excited electronic state such as that marked (2) in the same 
figure, then the energy of the quantum absorbed will normally 
lie between AD and BC. It is supposed that the atomic nuclei 
are vibrating slowly between A and J?, and that the change in 
the electronic configuration is rapid compared with the nuclear 
motion. 

It will be seen that the energy hv required to make an 
electron jump to an excited electronic state in a molecule is in 
general greater than the minimum energy required to reach 
the excited state. 



92 THE MANY-BODY PROBLEM 

Exercises 

(1) Prove that at temperatures such that kT$>hv the width 
(ED in Fig. 21) of the absorption band of a molecule corre- 
sponding to a given electronic transition is proportional to T*. 

(2) Obtain a solution of the Schrodinger equation for the 
potential energy function introduced by Morsef 



for the case I = 0. 

(3) How much does the value of / affect the equilibrium 
value R Q of R, and the vibrational frequency v ? (In the approxi- 
mation of 2-2 there is no change with I.) 

(4) If the potential energy function near the minimum is 

V(R) = -U + a(R- tf ) 2 + j8(/Z - ) 3 /? , 

use perturbation theory (Chap. IV, 9) to evaluate the effect 
of the final term on the energy of the first and second vibra- 
tional states. In practice how great do you expect this cor- 
rection to be ? 

3. SYSTEMS CONTAINING TWO OR MORE PARTICLES 
OF THE SAME TYPE 

In this section we shall discuss certain important properties 
of systems containing two or more particles of the same type. 
Examples of such systems are the helium atom (two electrons), 
the hydrogen molecule (two protons, as well as two electrons), 
the oxygen molecule (two oxygen nuclei). 

We shall begin by confining ourselves to systems containing 
two such particles. We shall first prove, as a theorem in mathe- 
matics, that if the system is in a non-degenerate stationary 
state with quantised energy, the wave function is either sym- 
metrical or antisymmetrical in the coordinates of the two 
particles. 

By this we mean the following. Let q 1 denote the spatial 
coordinates (x^y^z^ of one of the particles together with its 

f P. M. Morse, Phys. Rev. xxxiv, 57, 1929. 



SYSTEMS WITH PARTICLES OF SAME TYPE 93 

spin coordinate a v if it has a spin.f Similarly, let q 2 denote the 
coordinates of the other particle. Then the wave function des- 
cribing the two particles may be written 0(</i,</ 2 )- The wave 
function is said to bej symmetrical if 



and antisymmetrical if 



The proof is as follows. For brevity we write I for q^ 2 for q 2 . 
Tf the wave equation is written 

{fl(l,2)-W}0(l,2) = 0, (13) 

then the operator //(I, 2) is necessarily symmetrical; i.e. 

//(1,2) = //(2,1). (14) 

This follows from the fact that the particles are identical. 
Suppose, then, we interchange 1 and 2 in equation (13); we 
obtain 

(//(2,1)-W) 

and by (14) this may be w r ritten 



It follows that 0(2, 1) is a solution of the original wave equation 
(13). But we have already stated that 0(1,2) is a non- 
degenerate solution; that is to say, for the energy W there is 
no other solution satisfying the boundary conditions. Thus 
i/r(2, 1) must be a multiple of 0(1, 2), so that 

0(2,1) = 40(1. 2). (15) 

Interchanging 1 and 2, this gives 

0(1, 2) = 40(2,1). (16) 

I KIcctrons, protons, and neutrons have a spin, alpha-particles do not. 
| Thus the function 



/i-#2) 8 f (Zi-z*) 2 ) 

is symmetrical, cos0 = (2 1 ~2 2 )/r antisymmetricjil, and functions such as 
(1 -f*cos0)/r unsymmetrical. 



94 THE MANY-BODY PROBLEM 

Multiplying these equations together and dividing by 

0(1,2)0(2,1), 

we find A 2 = 1. 

whence A 1 . 

It follows that all non -degenerate quantised solutions are 
either symmetrical or antisym metrical. 

The simplest example is the idealised problem of a diatomic 
molecule rotating in two dimensions, i.e. in a plane; the rota- 
tion is described by the wave functions e ti0 , where 6 is the 
angle which the line joining the nuclei makes with a fixed axis. 
Since 6 changes to TT -f when the positions of the particles are 
interchanged, it will easily be seen that wave functions for 
even values of / are symmetrical, those for odd values anti- 
symmetrical. The same is true for the wave functions for 
rotation in three dimensions, -FJ M (cos 8) e iu $. 

We shall next prove that, if a system containing two identical 
particles is in a state described by a symmetrical wave function, 
it can never ma-ke a transition to a state described by an anti- 
symmetrical wave function, and vice versa. This follows at 
once from the wave equation which determines the rate of 
change of 0, namely (cf. p. 46) 

4f = //(!, ^(1,2). 

Since H(1 9 2) is symmetrical, the change 80 in which takes 
place in time 8t must have the same symmetry as 0. Thus a 
symmetrical function will stay symmetrical and an anti- 
symmetrical one will stay antisym metrical, for all time and 
under any perturbation whatever. 

We must now appeal to experiment, and state that for 
electrons, protons, and neutrons quantised states with anti- 
symmetrical wave functions are the only states found in 
nature, while for certain nuclei, notably He 4 , C 12 and O 10 , only 



SYSTEMS WITH PARTICLES OF SAME TYPE 95 

those with symmetrical wave functionsf are found. The evidence 
for this w r ill be given below; we discuss first, however, the reason 
why only half the theoretically possible states occur in nature. 
Suppose that two particles are under consideration, and that 
measurements are made of the position, momentum, and spin 
direction of one, which can be represented in the sense of 
Chapter III, 3, by a wave function w a (q), and also of the 
other, which can be represented by a wave function i(^(q). 
Then it might seem natural to set for the wave function 
f particles 



Then | w^qj w h (q 2 ) \*d^d< h 

would be equal to the chance of the first particle having co- 
ordinates between q and qi + dq 2 and the second between q 2 
and q 2 -f dq 2 . But such a wave function tells us more about the 
system than we can in fact know about it. If the particles are 
of the same type, it is impossible to tie a label on to one of 
them, and to say that this is the 'first particle 1 , and it is this 
one that we find at q v The wave function ^(q^q^) should give, 
when one writes down and interprets its square 1 | 2 , the chance 
that at q t one particle will be found, and at q% the other, without 
making any statement about which is which. If this is the 
correct interpretation of the wave function, it is clear that 



must be symmetrical. Further, this will be the case if we set 



but not for the simple products. Unless, however, we make some 
hypothesis as to whether the wave functions should be symme- 
trical or antisymmetrical, an ambiguity exists in w r ave mechan- 
ics; we do not know whether to take the plus or minus sign. 

f Particles for which the wave functions are symmetrical are said to obey 
Kinstejn-Bose statistics, those for which they are antisymmetrical Fermi- 
Dirac statistics. 



96 THE MANY-BODY PROBLEM 

4. APPLICATIONS OF THE SYMMETRY PROPERTIES 
OF THE WAVE FUNCTIONS 

4-1. The exclusion principle 

This important principle, first introduced into wave mechan- 
ics by Pauli, states that no two electrons in a given atom can 
have the same quantum numbers. The validity of this principle 
shows at once that wave functions describing electrons are anti- 
symmetrical. For suppose two electrons, to the approximation 
in which we can neglect their interaction, have the wave func- 
tions w a (q), w b (q). The antisymmetrical function formed from 
these is 



But if w a (q)> u} b (q) are the same function, as they will be if both 
electrons are in the same state, this must vanish. Therefore it 
is not possible for both particles to be in the same state, if the 
function is antisymmetrical. 

The wave function w includes the spin coordinate; a more 
detailed discussion of the spin coordinates is given in 4-3. 

4' 2. Particles without spin obeying Einstein-Bose statistics 

If a particle has no spin, it is described by the spatial or 
orbital coordinate r = (x, y, z) only. Certain nuclei such as He 4 , 
C 12 and O 16 , already mentioned as obeying Einstein-Bose sta- 
tistics, have no spin. We consider rotational states of mole- 
cules containing two of these nuclei (e.g. 2 ). The rotational 
wave functions of a diatomic molecule are symmetrical! in the 
coordinates r a , r 2 of the two nuclei for even values of the quantum 
number / describing the rotation of the molecule, antisym- 
metrical for odd values of /. Analysis of the band spectra shows 
that only states with even values of I are found for 2 and for 
the (unstable) molecule He 2 . This shows that the nuclei have 
no spin and obey Einstein-Bose statistics (i.e. have symmetrical 
wave functions). 

t This is true for the lowest electronic state; for certain excited states it 
may be the other way round, as may be seen from the arguments of 6-3. 



PARTICLES WITH SPIN 97 

Other evidence can be derived from the scattering of a-parti- 
cles in helium, where the symmetry of the wave function describ- 

ing the two particles leads to anomalies in the scattering, f 

f 

4*3. Particles with spin obeying Fermi-Dirac statistics 

It has been emphasised in Chapter IV, 13, that the effect 
of the spin on the orbital wave functions is small, and that to a 
good approximation one can represent the wave function of a 
particle with spin by a product of the form 



a is here + 1, according as the spin direction lies parallel or 
antiparallel to a fixed axis, and x capable of taking two 
independent forms x and xp> In the same way, then, the wave 
function of a pair of particles with spin can be written 



where <// is a solution of an equation of the type (3) in which 
the spin is neglected. The symbol r l here denotes x l9 y l9 z l9 etc. 
Now solutions of this equation, for non-degenerate quantised 
states, are either symmetrical or antisymmetrical in 14, r 2 ; we 
may write them ^^r^r^^^r^r^. Thus the wave function 
of the whole system, which must be antisymmetrical for an 
interchange of q 1 (denoting the whole group x lt y/ a , z l ,a 1 ) and # 2 , 
must have one or other of the forms 



where xs> XA are themselves symmetrical and antisymmetrical 
in oj, a 2 . Now such functions can be formed from x*>Xp only as 
follows: 

A symmetrical function xs( a v a z) can ^ e formed in three ways: 

X(l)Xa(2), 

(18) 



t Cf. Mott and Mussey, p. 102. 



98 THE MANY-BODY PROBLEM 

and an antisymmetrical function XA^V^) * n on ' v one way: 



We thus reach the following conclusion. In any system con- 
taining two particles with spin, the states can be separated into 
those with symmetrical orbital wave functions i/j s , including 
usually the ground state, and those with antisymmetrical orbital 
functions ^ A . Transitions between states w r ith symmetrical and 
antisymmetrical orbital wave functions, though possible, will 
have very low probability; to the approximation that the wave 
function (17) is valid, they do not occur. All states with sym- 
metrical orbital wave functions are singlet states, the two spins 
being antiparallel so that they contribute nothing to the 
mechanical or magnetic moment; all states with antisym- 
metrical orbital functions are triplets, the total spin moment 
along a fixed direction being, in multiples of ft, 1, 0, or -f 1. 

4-4. The rotational states of H 2 

The proton has a spin with angular momentum &H and a 
magnetic moment of about two nuclear magnetons (^Hj2Mc). 
Owing to the smallness of this magnetic moment, which will 
influence the orbital wave functions very little, the transition 
probabilities from rotational states for which / is even (sym- 
metrical wave functions) to those for which / is odd will be 
very small indeed. In fact for ordinary purposes hydrogen gas 
can be regarded as a mixture of two gases, parahydrogen (mole- 
cules in singlet states for which / is even) and orthohydrogen 
(molecules in triplet states for which I is odd). Transitions from 
one state to the other take place practically only in the presence 
of a catalyst which dissociates the molecules into atoms, so that 
an atom from one molecule can recombine with that from 
another, f In the absence of a catalyst, when the gas is cooled 
the numbers of molecules in the various rotational states will 
not reach the equilibrium values, because molecules cannot 

fCf. A. Farkas and L. Farkas, Proc. Hot/. Soc. A, CMI. 124, 1985; D. D. 
Eley, 'The Catalytic Activation of Hydrogen', contribution to AfhwHres in 
Catalysis, New York, 104?, vol. 1, p. 157. 



HELIUM ATOMS 09 

jump from the state / = 1 to I = 0. This fact has an important 
effect on the specific heat at low temperatures, which has been 
observed experimentally.! 

4-5. The helium atom 

The Schrodinger equation for two electrons moving in the 
field of a nucleus is 

= 0, 



where //0 = --(Vf + V|)0+ '0, (19) 

~*iiL 

2 2 2 

and where 



The first two terms in F represent the potential energy of the 
two electrons in the field of the nucleus, and the final term the 
interaction energy of the two electrons. 

It follows from the arguments of 4-3 that the solutions of 
this equation are either symmetrical or antisymmetrical: that 
states with symmetrical orbital functions are singlets (known 
as parahelium states), that states with antisymmetrical orbital 
functions are triplets (known as orthohelium), and that transi- 
tions from one series to another have very low probability. In 
this section we shall show how to calculate approximately the 
energies of these states. 

We shall start with the approximation in which each electron 
is given a separate orbital wave function, denoted by *f* a (r), 
*fj b (r). We shall suppose, moreover, that these functions are 
either the same (ift a = i// b ), or else that they are orthogonal. 
Then an approximate wave function describing the pair of 
electrons will be 

0(1,2) = ^{0 rl (l)^(2)0 rf (2)0 6 (l)}. (20) 

t Cf. the review by R. II. Fouler. Statistical Mechanics* *Jnd ed., Cam 
bridge, 193(i, p. 82. 



100 THE MANY-BODY PKOBLEM 

The factor 1/^2, if ifj (l and ^ ft are normalised, ensures that the 
wave function ^(1,2) is normalised, in other words that it satis- 
fies the equation 



Jj 



= 1. 



Approximate values of the energy of the atom can then be 
found from the formula 



W being here the total energy of both electrons. 
Substituting for t^(l, 2) we find 

W = IJ, (21) 



where ,.., 

(22) 



The integral J is known as an 'exchange' integral. 

On the basis of this theory we can give a descriptive analysis 
of the level scheme of helium: for the ground state we may set 
ifj a = ijj bJ both being spherically symmetrical functions of the 
type Ce~ rla . Then only a symmetrical function of the type 
0( r i)0( r 2) 1S possible. Bu if one of the electrons is excited, 
there exist two states with different energy levels, corresponding 
to the two signs in (20) and (21). Moreover the ground state 
and the excited states which have symmetrical wave functions 
in the spatial coordinates r l5 r 2 are singlet states (known as 
states of parahelium); the spin wave function is of the form 



and so the two spins point in opposite directions and make no 
contribution to the mechanical and magnetic moments. The 
ground state, for which both electrons have spherically sym- 
metrical wave functions, has thus no resultant angular mo- 
mentum or magnetic moment at all; excited states, such as 



HELIUM ATOMS 



101 



that in which one electron remains in the Is state while the 
other is in a state with orbital angular momentum IH, have 
angular momentum of this amount and magnetic moment 
p, B L They show a normal Zeeman effect. 

The states with antisymmetrical orbital wave functions 
(states of orthohelium) have the three spin wave functions 



PARAHELIUM ORTHOHELIUM 



3Si 





25, 



Metastable 
level 



Fig. 22. Energy levels of helium. 

given by (18). They are thus triplet states, of total spin 
moment one unit of //. Provided that the excited electron 
is not in an s state (/ = 0), the spin magnetic moment will 
interact with the orbital moment so that the state will split 
into three states with different energies. States in which the 
excited electron is in an s state will not be split except in the 
presence of a magnetic field. 

A schematic representation of the energy levels of helium is 
shown in Fig. 22. 



102 THE MANY-BODY PROBLEM 

We may note heref that the exchange integral is positive. 
Thus the triplet levels (orthohelium) lie below the singlet levels. 
This may be understood, because an antisymmetrical orbital 
wave function must vanish when tj = r 2 and will in general be 
small when the particles are close together. Thus the positive 
contribution to the energy made by the interaction term 
e 2 / 1 r 1 r 2 1 is smaller than for the symmetrical states. 

4 6. The structure of atoms iviih more than two electrons 

In describing atoms more complicated than helium, the most 

convenient approximation is the following: 

All electrons are supposed to move in the so-called self- 

consistent field. This field, in which the potential energy of an 

electron is V(r) y is defined as follows. For small r 



where Z is the atomic number, and for large values of r 



At other points it is defined as the field of the nucleus and the 
average field produced by all the other electrons, if each of them 
is treated as producing a charge density e \ $(r) | 2 . It will be 
noticed that with this definition V(r) is not quite the same for 
each electron. 

In this field the lowest level, with principal quantum number 
758. = 1, is known as the K level. The energy of such a level is 
given approximately by the Bohr formula 



t For the evaluation of the exchange integrals and suitable choice of the 
wave functions fa, fa, the reader is referred to the following authorities: 

(i) The original paper on the subject is by W. Heisenberg, Z. Phys. xxxix, 
409, 1926. 

(ii) A very complete account of the calculations made up to 1938 is by 
H. Bethe, Handb. Phys. xxiv, pt. 1, 342, 1933. 

(iii) L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, 
New York, 1935, p. 210. 



ATOMS WITH MORE THAN TWO ELECTRONS 



103 



for Z = 79 (gold), for instance, this is 84,000 eV. By the Pauli 
principle two electrons can be accommodated in the K level. 

The next set of levels, with principal quantum number 2, can 
accommodate eight electrons, two in s states (I = 0) and six in 
p states (I = 1). The electrons in these states form what is 
known as the L shell. There are in fact three L levels; the level 




o 0-5 i-o o 1-5 2-0 

Atom radius r in A 

Fig. *23. Radial charge distribution for the different electron 
groups of K + . 

Lj, with quantum number / = (2s), which can accommodate 
two electrons, and the levels L fl and L in , with 1=1 (2p), and 
spin directions either parallel or antiparallel to the direction of 
the orbital momentum. 

Owing to the Pauli principle, an electron cannot normally 
jump from an L to a K level, since all the states in the K level 
are occupied. Under electron bombardment, however, a K 
electron may be ejected from an atom; an L electron may then 
fall into the vacant level, a quantum of X-radiation being 
emitted. Processes of this type are responsible for the emission 



104 THE MANY-BODY PROBLEM 

of X-ray lines. The transitions from the L u and L UI levels to 
the K level give rise to the lines known as K ai and K^. 

Fig. 23 shows the charge distribution calculated by the 
method of the self-consistent field in the several shells of the 
potassium positive ion. The quantity plotted here is r 2 |</f| 2 , 
and thus the number of electrons between spheres of radii 
r, r-f dr, divided by rrdr. 

Some points to notice about the shell distribution of electrons 
in atoms are the following: 

(1) Normally only the outermost shell of electrons in an 
atom is responsible for chemical binding, and is thus affected 
by the chemical state of an atom. Therefore the frequencies 
and breadths of most X-ray emission lines, unlike optical 
spectra, are almost unaffected by the state of chemical binding. 
Breadths and structures of lines which start from the outer- 
most level do, however, depend on the chemical state of the 
atom (cf. 7). 

(2) Any closed shell, K, L or M, i.e. one in which all the 
states of a given principal quantum number are occupied, has 
a spherically symmetrical charge distribution, and no resultant 
spin or magnetic moment. 

The reader will easily verify that for an L shell, for instance, 
the distribution is spherically symmetrical; since the three p 
functions have the form 

*/(r), Vf(r), *f(r), 
the resultant charge density is 



(3) Information about the charge density in atoms can be 
obtained experimentally from the intensities with which atoms 
(for example in a crystal) scatter X-rays. The argument is as 
follows. If a polarised light wave falls on a free electron, the 
classical formula for the scattered amplitude, measured at a 
point P at distance R from it, is| 

E e 2 . . 



t Cf. Chap. IV, 10. 



ATOMS WITH MORE THAN TWO ELECTRONS 105 

here <f> is the angle between the electric vector E of the incident 
wave and the line joining the electron to the point P. This 
formula is not in fact valid for free electrons, for which the 
Compton effect with recoil of the electron is to be expected; 
but the formula may be used to calculate the coherent scattering 
by an atom of radiation for which the frequency is great com- 
pared with the absorption frequencies of the atom. One assumes 
that if e P(r) is the charge density, then each element of volume 
dr scatters a wavelet of amplitude 



and that these wavelets interfere. The resultant amplitude 
may be calculated exactly as in Chapter II, 7; corresponding 
to formula (11) the resultant amplitude is 

n^ (23) 



i_ ET/m f aj r>/ , sin(47rrsin0/A) 

where F(0) = P(r) ,~ --TT/Y 

Jo 4T7/- sin 0/A 

20 is here the angle of scattering and A the wavelength of the 
X -radiation.*)* F(6) is known as the atomic scattering factor. 
For an unpolarised wave sin <f> must be replaced by 



(4) The diamagnetic susceptibility depends critically on the 
radial extent of the wave function; it is given byj 



Ne* C 

= 2) S^l 



I For a review of this subject, see, for example, R. W. James, The. Optical 
Principles of the Diffraction of X-ray ft t London, 1948, chup. HI. For some 
recent determinations of the electron density in the ions of alkali -halide 
crystals, see R. Brill, H. G. Grimm, C. Hermann and C. Peters, Ann. Phys., 
Lpz., xxxiv, 393, 1939, or the review of this work by A. Eucken. Lehrbuch 
der chemischen Physik, 1944, vol. n, pt. 2, p. 537. 

J Cf. E. C. Stoner, Magnetism and Matter, Leipzig, p. 108. 



106 THE MANY-BODY PROBLEM 

where N is the number of atoms per unit volume, and the 
summation is over all the n electrons of the atom. 

A number of approximate methods exist for obtaining wave 
functions of atoms. There is first of all the method of the self- 
consistent field due to Hartree, already mentioned. Then 
various improvements to the Hartree method, notably that 
of Fock, have been developed. f Finally the Thomas-FermiJ 
method is available for atoms too complicated to be treated 
by other methods. 

5. INTERATOMIC FORCES AND THE FORMATION 
OF MOLECULES 

Forces between atoms are of the following types: 
(a) Van der Wools attractive forces.^ A calculation based on 
wave mechanics shows that, at sufficiently large distances r, all 
atoms and molecules attract each other with a force of which 
the potential energy V(r) is of the form <7/r 6 . This force is of 
importance in considering the equation of state of imperfect 
gases, and is responsible for cohesion in solid or liquid rare 
gases, methane (CH 4 ), solid hydrogen, etc. The constant C is 
given in terms of the absorption frequencies v r and the related 
oscillator strengths f r . 

If we make the approximation that the oscillator strengths 
of all lines can be neglected except those of one line for each 
atom, of frequencies v, v , then 



(f 2 \ 2 1 
__r I 
47T 2 TW / in/ (v -f v' ) 



For the rare gases, where no other attractive force comes into 
play, the van der Waals attraction is rather weak, on account 
of the high values of the excitation potentials. This is shown 
by the low values of the melting and boiling points of these 
substances. 

f A review of these methods and of the results obtained have been given 
by D. R. Hartree, Rep. Prog. Phy*. n, 118, 1040-7. 
t Cf., for example, Mott and Sneddon, p. 158. 
For further details compare, for example, Mott and Sneddon, p. 14. 



COVALEXT FORCES 107 

The van cler Waals force is the only force between neutral 
atoms when their charge clouds do not overlap. 

(6) Repulsive overlap forces. As soon as two closed shells 
overlap, a strong repulsive force sets in; in principle this can 
be calculated by the methods of wave mechanics, but the 
calculations are laborious and the results obtained not always 
reliable ;f it is usual to use empirical forms such as 

Ar~* (,9-9-12), or Be^ r 

for the potential energy of two atoms distant r apart. 

(c) Ionic forces. It seems to be a good approximation to treat 
alkali-halide crystals as made up of positive metal ions and 
negative halide ions, the crystal being held together by the 
electrostatic attraction between them, and the ions kept apart 
by the repulsive overlap forces. Much work has been done in 
explaining in terms of these forces the properties of crystals 
built of atoms or ions of which the outermost electronic shell 
is closed; e.g. alkali-halides and solid rare gases. This, however, 
is not strictly a part of wave mechanics, and we shall confine 
ourselves to giving references here.J 

6. COVALENT FORCES 

Apart from the van der Waals forces and electrostatic attrac- 
tion between ions, forces between atoms arise only when the 
wave function of one atom overlaps that of the next. The 
valence forces of chemistry are of this type. 

f For a recent attempt, cf. G. VVyllie and E. F. Benson, Proc. phys. Soc. A, 
LXIV, 276, 1U31. 

J For alkali-halides and crystals held together by ionic forces, see the 
following: M. Horn and M. Goppert-Mayer, Jlandb. Phys. xxiv, pt. 2, G25, 
11)83; N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, 
1940, chap, i; .1. Sherman, Chem. Rev. xi. 153, 11)32. 

For criticism of the approach under (c) above, and the refinement? intro- 
duced by wave mechanics, see P. O. Lowdin, ^4 Theoretical Investigation into 
some Properties of Ionic Crystals, Uysala, 1940. 

For a discussion of solid rare gases, see J. K. Lennurd- Jones and B. M. 
Dent, Proc. Roy. Soc. A, cxm, 073, 1927. 



108 THE MANY-BODY PROBLEM 

We shall discuss valence forces in the following way: 
(a) We shall give a treatment of the movement of an electron 
in the hydrogen molecular ion H 2 +, where the problem is 
that of the movement of a single electron in the field of two 
protons. 

(6) We shall discuss the hydrogen molecule H 2 by a method 
known as the method of molecular orbitals,f in which each 
electron is pictured as shared between the two atoms in the 
same way as in H^, so that each electron can move inde- 
pendently from one atom to the other. 

(c) An outline will be given of an alternative mathematical 
approach, that of London-Heitler, in which each electron is 
located on its own atom; they are allowed to change places, 
but not to move across independently of each other. 

It will be realised that (6) and (c) are alternative approxima- 
tions; the true wave function will lie somewhere between the 
two extremes. 

(d) Finally we shall enumerate some of the main points in 
the application of wave mechanics to more complicated 
molecules. 

6-1. The molecular ion H 2 + 

The essential point in the treatment of this ion is that for 
every state of the electron in a single atom there will be two 
states for the molecule. Consider, for example, the Is state of 
the atom, with a normalised function of the form 



where r is the distance from the nucleus. If the two nuclei are 
at points defined by the vectors 



t The name is due to J. E. Lennard-Jones, Trans. Faraday Soc. xxv, 
8, 1929. 



HYDROGEN MOLECULAR ION 109 

then I r - r tt | = V((* - *,) 2 + (.'/ - </) 2 + (* - s) 8 } 

represents the distance of the electron from the point r a ; a 
similar expression may be written down for the distance from 
r b . Then Ave may write 

a (r) = Mr-r a |) 
for an electron in the ground state in atom a, and 



for an electron in the ground state in atom 6. 

An electron in the molecule may be located on either of the 
atoms; one may write the wave function 



By symmetry it is just as likely to be on one atom as on the 
other; that is A 2 must be equal to J5 2 , so that A = B and the 
two possible wave functions (unnormalised) are 



and H^ = ^(r)-0 6 (r). (24-2) 

Moreover, these have different energies; using the formula for 
the energy W from Chapter TV, (13), in the form 

W f| T \ 2 dr ---- | V F*//WT, (25) 

we see that W( 1 A) = / J, 

where A = 



J = 



110 



THE MANY-BODY PROBLEM 



Exercise 

Show that A 
the energy W. 



B by minimising the expression (25) for 



We shall not discuss the numerical evaluation of the integrals 
A, /, J, since formula (25) gives only an approximate form for 
the wave function of H 2 + ; actually it is possible to obtain an 
exact solution of the Schrodinger equation for an electron 
moving in the field of two nuclei. f For our purpose, however. 

5 IP/? (atomic 

i units) 




-0-5 
W 

-10 
-15 

-2-0 

Fi#. 24. Energies of an electron in the states of (1) even and 
(2) odd parity of the ion // 3 +. The full lines represent the energy of 
the electron, the dotted lines with the addition of the term c 2 /K. 

we need only notice that the symmetrical solution has no 
nodal surface; the energy of the electron will tend to W as 
the distance K between the nuclei tends to infinity, and 4H^ 
as R tends to zero. W Q is here the ionisation potential of 
hydrogen, we 4 /27i 2 . The electron's energy W(R) thus exerts an 
attractive force between the two nuclei ; and it is not surprising 

t Cf. O. Burrau, K. Danske Vidcnsk. Sehk. vn, 14, 1927; K. A. Hylleraas, 
Z. Phys. LXXI, 780, 1931; G. Steensholt, 2. Phytt. c, 547, UWC. 



HYDROGEN MOLECULAR ION 111 

that when the potential energy of the nuclei in one another's 
field is added to give the total energy V(R), 



one obtains a curve with a minimum (Fig. 24). On the other 
hand the state (24-2) with odd parity tends, as R tends to zero, 
to the 2p state of an electron in the field of a charge 2e, which 
has energy W Q . The electron thus exerts no attractive force; 
and when the term e 2 /R is added the potential energy leads to 
repulsion only. 

l/ 



Fig. 25. Wave functions of the hydrogen molecular ion; (0), (6), 
and (c) are plotted along the line joining the nuclei; (a) and (6) show 
the two states derived from * atomic wave function, (c) and (d) 
those derived from p atomic wave function. 

The considerations of Chapter VI show that optical transi- 
tions are allowed between the two states considered here. 

The states formed from atomic wave functions of p sym- 
metry are also of interest. If the nodal surface is perpendicular 
to the line joining the nuclei, the wave function corresponding 
to the lower of the two states will be as in Fig. 25c. If, on the 
other hand, the nodal surfaces are in the plane of the paper, 
the wave function will change sign in going through the paper, 
but when they are plotted along any line parallel to the line 
joining the nuclei the wave function will be as in Fig. 25d. 
States for which a nodal surface passes through the line joining 
the nuclei are called TT states; those for which the line joining 
the nuclei contains no node are called a states. 



112 THE MANY-BODY PROBLEM 

6-2. The method of molecular orbitals for the hydrogen mole- 

cule H 2 

In this method the two electrons in the molecule are both 
described by the same wave function T(r), namely, one of the 
type (24* 1) illustrated in Fig. 25a. The wave function of the 
pair of electrons in the ground state is, including spin co- 
ordinates. 



The ground state, like the ground state of the helium atom, 
thus has no magnetic moment due to its electrons. 

Excited states can be treated in a similar way. As for helium, 
there is a singlet and a triplet series of terms. t 

The error in the method of molecular orbitals is that it 
neglects altogether any correlation between the positions of the 
electrons; it suggests that it is as likely that both electrons are 
located on the same atom as that they are on different atoms. 

ti-3. The method of London -Hei tier as applied to H 2 

This method of approach goes to the opposite extreme, and 
starts with an approximation in which the two electrons are 
located definitely on different atoms. The electrons are allowed 
to change places, but not to move independently from atom 
to atom. 

The wave function that we use must be antisymmetrical in 
bhe coordinates of the two electrons, including spin. The two 
possibilities, starting with Is wave functions ifj a (r),ifj b (r) for an 
electron in either atom, are 



) ) (26) 

). (27) 

The former is the wave function of the ground state. Here the 
intisymmetrical spin wave function, as for the helium atom, 
*ives a singlet state, with zero spin moment. 

t It rnu*.t be emphasised that each electronic state is split into a very 
afge number of states by the quantised rotational and vibrational states 
liscussed in 2-2 above. 



CHEMICAL BINDING 113 

The energy W of the two states is, as in (21) and (22), 

(7J)/(1A), 
where /,/ are formally as defined on p. 100 and 

A = 



The wave functions are now not orthogonal and A does not 
vanish, as it does for the analogous case of the helium atom. 
The integrals can be worked out.f The exchange integral J is 
negative, and, together with the energy e 2 /R of the Coulomb 
interaction between the nuclei, leads to attraction between the 
atoms for state (26), repulsion for (27). 

In the absence of overlap between the wave functions, / is 
just equal to the energy of a pair of free atoms, and J and A 
vanish. Cohesion of the type described here is a result of over- 
lap, and in general maximum cohesion occurs when the overlap 
is a maximum. 

6-4. Some general features of chemical binding 
The considerations of the previous section show 
(a) that a bond involving two electrons, one from each atom, 
can only be formed if the two spins are antiparallel; 

(6) that a strong bond involves considerable overlap between 
the wave functions. 

We shall apply these principles first to the water molecule 
H 2 O. The oxygen atom has two electrons in the state 2s and 
four in the state 2p. Thus two of the 2p wave functions are 
necessarily paired already, while two are ready to form bonds 
with hydrogen. But these two must be different wave func- 
tions; they must necessarily have their nodal surfaces at right 
angles to each other. That is why the lines joining the nuclei 
of the two hydrogen nuclei to the oxygen nucleus make approxi- 
mately a right angle with each other (actually 105). 

t The original paper is that by W. Heitler and F. London, Z. Phys. xuv, 455, 
1927. See also reviews of later work by H. Bethe, fiandb. Phys. xxiv, pt. 1, 
535, 193.3; and Pauling and Wilson, Introduction to Quantum Mechanics, p. 340. 

8 



114 THE MANY-BODY PROBLEM 

Another important feature of the theory of chemical binding 
is the occurrence of hybridised wave functions. An example is 
the behaviour of carbon in diamond or in such compounds as 
CH 4 . Here each carbon atom is surrounded at the four corners 
of a tetrahedron by atoms with which it forms a homopolar 
bond. It would not be correct to say of the four outer electrons 
in carbon that two are in s states and two in p states; it is 
correct to ascribe to each of them one of four 'hybridised' wave 
functions, which extend as far as possible in the four tetra- 
hedral directions, so as to give the maximum overlap. These 
four wave functions are 



(28) 



The student is recommended to verify that they have the 
desired properties. 

7. THE THEORY OF SOLIDS 

In this section there will be space only to summarise a few 
of the more important contributions made by wave mechanics 
to the theory of solids. 

7-1. Concept of a conduction band 

Consider any normally non-conducting crystal as, for 
example, sodium chloride. Suppose that an extra electron is 
brought from outside and placed on one of the metal ions Na + . 
Then according to wave mechanics such an electron will have 
the following property. It will not stay localised on one ion, 
but will be able to jump freely from one ion to the next. 
Moreover, in a perfect lattice, if an electric field F is applied, 
the electron will be accelerated, the acceleration being given by 



CONDUCTION BAND JN SOLIDS 115 

m eff , the so-called effective mass, may differ somewhat from 
the mass of a free electron, but is of the same order of magni- 
tude. The concept of a mean free path does not arise for an 
idealised lattice in which all the ions are held rigidly in position. 
This property of an extra electron brought into a crystal can 
be deduced from the following properties of the solutions of 
the Schrodinger equation for an electron moving in a periodic 
field. Let V(x,y,z) be the potential energy of an electron 
moving in the field that it will encounter within the crystal. 
Then V(x,y,z) will have the same period a as the crystal 
lattice. The Schrodinger equation is 



, 

The solutions may be shown to have the form 

</, = ^X(#,//,2), (29) 

where u k (x,y,z) is periodic, with the same period as the lattice. 
This may be compared with the form 



if, = 

for a free electron. Thus the wave function of an electron in 
the lattice is similar to that of a free electron, and represents a 
particle moving in a definite direction without being scattered. 
It is modulated, however, by the field of the lattice. 

A mean free path arises if the electrons are scattered, and 
this occurs only if atoms are displaced from their mean position 
by heat motion, or if impurities are present. In either case the 
electron wave is scattered. In the case of an atom displaced 
from its mean position a distance x, the amplitude of the 
scattered wave is proportional to x, and the intensity is pro- 
portional to x 2 . Since the mean value of ar 2 varies as the tem- 
perature, so does the resistance of a metal. 

Corresponding to each wave number k there is an energy 
value W(k). For small k, W is of the form 

W(k) = 



116 THE MANY-BODY PROBLEM 

though for higher energies it is more complicated, and bands 
of forbidden energy occur. The first band of allowed energies 
is known as the 'conduction band'. 

7-2. Semi~condwtors 

Semi-conductors are substances for which the conductivity 
increases with increasing temperature. Most but not all are 
activated by impurities; that is to say, their conductivity 
depends on the presence of small traces of impurity present in 
concentrations from one part in 10 6 to about one per cent. It is 
thought that these impurities are dispersed in atomic form, and 
that each such impurity centre can release an electron with the 
expenditure of an amount of energy W that is not too big 
compared with kT, and is small compared with the excitation 
energy of a free atom. The electrons released are, as we say, 
'in the conduction band'; they are free to move through the 
lattice. 

If there are N impurity centres per unit volume, it can be 
shown that in thermal equilibrium the number n of free elec- 
trons (electrons in the conduction band) is per unit volume 



where N = (2TrmkT/h 2 )*. 

The conductivity a is obtained by multiplying by ev t where v 

is the mobility, 

J a = nev. 

Both N and v vary with temperature, but frequently the 
exponential is the predominating term, so that the slope of a 
plot of Ino- against 1/T enables an estimate of W to be made. 

Actually the slope of such a curve decreases with increasing 
concentration of impurity, f for a reason which is not yet fully 
understood. 

The current theory of the nature of the impurity centres 
which explains qualitatively why the values of W are so much 

f Cf. The Report on the Heading Conference on Semi -conductors, London, 
1951. 



SEMI-CONDUCTORS 117 

smaller than the ionisation potentials of free atoms, is as fol- 
lows. An impurity atom can be accepted by a non-metallic 
substance in various ways. For instance, ZnO accepts excess 
zinc by taking a zinc ion (Zn 4 ) into a so-called interstitial 
position, which means into one of the small gaps between the 
zinc and oxygen ions of the crystal lattice. Alkali-halides 
accept excess metal through the presence of lattice sites from 
which the anion (the halogen ion) is absent. In either case the 
centre carries a positive charge, so that the field round it is 
e/Kr 2 9 where K is the dielectric constant of the medium. The 
centre has to be neutralised by an electron. If an electron is 
'in the conduction band', that is to say, free to move from 
atom to atom, its potential energy in the field of the centre is 
e 2 /i<r. It can thus be held in quantised energy levels, exactly 
similar to those of an electron in the field of a proton, except 
that in all formula e 2 must be replaced by e 2 /* and m by m etf . 
The binding energy is thus 



m 

For materials such as silicon (*~17), activated by suitable 
impurities, this gives, assuming m eff /ra~l, about 0-01 Ve, in 
good agreement with experiment.")" 

If the wave function of the bound electron is of the form 
(7e~ r/a , the quantity a may be defined as the 'radius' of the 
orbital. It should be given by 

a~i<n 2 /me* = K x 0-54,4, (31) 

or about 1A for silicon. The impurity atoms are thus swollen 
owing to their presence in the dielectric. 

It is not suggested that these formulae, (30) and (31), are 
exact since the assumption that the potential is e 2 //cr cannot 
be true right up to the dissolved ion. However, they give 
correctly the order of magnitude. 

f Cf. G. L. Pearson and J. Bardeen, Phys. Itev. LXX\ , 863, 1949. 



118 



THE MANY-BODY PROBLEM 



7-3. Metals 

These are treated by a method similar to the method of 
molecular orbitals. We shall illustrate this by considering a 
'one-dimensional' metal, i.e. a row of N atoms, each containing 
one electron. It is convenient to think of this row as bent into 
a closed loop (Fig. 26), round which the electrons can move. 

We then ascribe to each electron a 
wave function of the type 

e***u k (x), (32) 

where x denotes the distance of the 
point considered from a fixed point 
on the loop. In order that the wave 
function (32) may join up smoothly 
again at 0, k must satisfy the quan- 
tum condition 




where L is the circumference and n is an integer. We now intro- 
duce the Pauli principle, that only two electrons, with spins in 
opposite directions, can be in states described by the wave func- 
tion with given k. Thus states are occupied with all integral values 
of n between \N; higher states, at the absolute zero of temper- 
ature, are empty. The electrons have kinetic energies between 
zero and H^ ai = 7r 2 H 2 (NjL) 2 ISm. Since (N/L) is the interatomic 
distance, this energy is of the order of several electron volts. 

The important conclusion of this calculation is that the 
electrons in a metal at the absolute zero of temperature are not 
at rest, but have energies between zero and some value W mta ^ 
of this order, and hence large compared with kT. This band 
of energy levels is known as the Fermi distribution of levels. 

The calculation can be extended to three dimensions; the 
formula for W is then 



_ 

a " - 2m 
where N is the number of electrons per unit volume in the metal. 



METALS 119 

It will be seen that, since H; nax ^> kT, raising the temperature 
will excite only a fraction of the total number of electrons of 
order kT/W m&x , and that these will each acquire energy of the 
order kT. Thus the total thermal energy of the N electrons 
in a metal is a numerical constant multiplied by 



The specific heat per electron is thus a multiple of 



and thus linear in the temperature. 

The electronic specific heat is important at low temperatures, 
since that due to lattice vibrations varies as T 3 . It has been 
observed at the temperature of liquid helium for a number of 
metals. | 

An exact discussion of the specific heat involves a treatment 
of Fermi -Dirac statistics, the statistics obeyed by particles 
such as electrons for which wave-functions must be anti- 
symmetrical. For this the reader is referred to text-books on 
statistical mechanics. 

The most direct experimental proof of the existence of a 
broad band of energy levels is provided by the form of the 
X-ray emission bands which result when an electron makes a 
transition from the conduction band to an X-ray level (Fig. 
27a). The width of the band gives directly the width of the 
band of occupied levels.J 

We show in Fig. 276 the usual energy level scheme for a 
metal; the band of occupied levels of width W m ^ (the Fermi 
distribution) and the work function <f> representing the minimum 
energy required to remove an electron from the metal. This 
can be determined either photoelectrically or from thermionic 
emission. 

t Cf. N. F. Mott and H. Jones, Theory of the Properties of Metato and 
Allmjft, 1936, pp. 182, 193. 

} For a review of this subject from the experimental point of view see, 
for example, II. W. H. Skinner, Rep. Prog. Phys. v, 271, 1938. 



120 



THE MANY-BODY PROBLEM 



Fig. 27c shows the energies in the presence of a field F 
pulling electrons away from the surface. OA represents the 
potential energy eFx of an electron in the field. If the field is 
strong enough, electrons can be pulled out of the metal through 



zero 




Fig. 27. Energy levels of electrons in metals, (a) X-ray emission; 
(6) Surface of a metal; (c) Strong field emission. 

the potential barrier BOA, by 'tunnel effect'. The calculation 
of the chance of penetration through the barrier can be made 
by the method of Chapter II, 6; the chance, that an electron 
with energy W mAX incident on the barrier will penetrate it, is 
e-t, where 



The current emitted per unit time is obtained by multiplying 
this by an approximately constant factor. One thus obtains the 
result that the current depends on the field F through a formula 

of the type , 

* r current = const. e~- 



where F 9 is a constant. 



CHAPTER VI 
TRANSITION PROBABILITIES 

1. GENERAL PRINCIPLES 

The problem to be treated in this chapter is the following. An 
atom is originally in a given stationary state, for example, the 
ground state. It is then perturbed by a passing charged par- 
ticle, by a light wave or in some other way. After the particle 
has passed, or after the light wave has irradiated the atom for 
a certain time, the atom may have made a transition to one of 
the other stationary states. Wave mechanics enables us to 
calculate the probability P that this has occurred. 

The principles by which such a calculation can be made are 
as follows. We denote the coordinates of the electron or elec- 
trons in the atom by g, and write the Schrodinger equation in 

* form (H -WMq) - 0, 

so that the quantities W n are the energies that the atom can 
have, and $ n (q) is the wave function describing the atom when 
the energy is W n . Introducing the time factor, the complete 
wave function is 



This satisfies the time-dependent wave equation 

* W 

i~st 

of which the general solution is 



(i) 



where the A n are arbitrary constants. Since, however, in the 
problem to be considered the atom is initially in the ground 
state, we must take for the initial form of the wave function 

(2) 



122 TRANSITION PROBABILITIES 

We then introduce the potential energy of the electrons in 
the perturbing field, for example, that of the passing particle 
or light wave. Explicit forms are given in (7) and (22). The 
potential energy will vary with the time; we write it V(q\ t). The 
wave equation for the electrons in the atom then becomes 



. (3) 

This equation, being linear in the time, serves to define *F at 
all subsequent times, since we are given the initial form X F of 
*F by (2). We may expand this wave function in the form (1) 
as indeed we may expand any arbitrary function of q in a 
series of the characteristic functions *ft n (q) of the unperturbed 
atom; but the coefficients A n will now be functions of the time. 
We therefore write the expansion of the wave function Y at 
time t in the form 



The coefficients A n (t) may be calculated; this will be done in 
the next paragraph. First, however, we are concerned with 
their interpretation. Their interpretation a new physical 
assumption mentioned already in Chapter III of this book 
is that | A n (t) | 2 is the probability at time t that, under the 
influence of the perturbation, the atom is in the stationary 
state n. A wave function such as (1) or (4), in which all or 
several normal modes are excited, describes a state of affairs 
in which we do not know in which state the atom is; we are 
only given the probability that it is in one state or another. 

Exercise 

Prove that ] | A n (t) | 2 does not vary with the time. 

n 

We shall now show how to calculate the coefficients. If we 
substitute (4) into (3), we obtain 



EXCITATION BY A PASSING PARTICLE 123 

and hence, multiplying by ifi*t ili l/h , and integrating over all 
space, we find 



Everything in the right-hand side is known except, of course, X F. 
An approximate solution may be obtained if we assume that 
the effect of the perturbing field is so small in the time con- 
sidered that all the coefficients A n remain small compared with 
A , so that the probability of excitation is small. In this case 
we may replace T in the right hand of (5) by its original form 
*F ft ; we have therefore 



JA ~~ *^ |Vn r V0""f (") 

where OJ I(Q = (W n W^IU. 

The value of A Jt at time t can be obtained by integrating from 
to t; \A n \ 2 then gives the probability that at time t the 
atom is in the state n. 

The approximation used here should give accurate results 
for the excitation of an atom by a light wave; one can always 
take the time short enough for all the coefficients A n to be 
small compared with A , unless the intensity of radiation is 
so great that the probability of excitation is no longer pro- 
portional to E 2 . For the case of a passing particle, the approxi- 
mation will no longer be good for close collisions. 

2. EXCITATION OF AN ATOM BY A PASSING PARTICLE 
As an example we shall work out in this section the proba- 
bility of excitation of a hydrogen atom by a passing particle, 
such as a proton. A proper treatment"!" describes the incident 
particle as well as the electron by a wave function; but the 
method given here, in which the proton is treated as a moving 
centre of force, does in fact give the correct answer if the mass 
of the particle is great compared with that of the electron. 

t Cf. Mott arid Massey. 



124 



TRANSITION PROB ABILITIES 



In Fig. 28 the atom is situated at the point 0, the particle 
moves along the line AB and passes the point C at the time 
t = 0. At time t y then, its coordinates are (vt,p, 0). Here v is 
the velocity of the particle, and p the 'impact parameter' or 




Fig. 28. Showing the excitation of a hydrogen atom at O by a 
passing proton moving along the line AB. 

distance between the path of the particle and the nucleus of 
the atom. The perturbing energy V is thus the potential energy 
rf an electron at the point (x, y, z) in the field of the proton, so 
that 

e 2 

. i -. (7) 



For the purpose of this example we shall consider only distant 
collisions, so that p is large in comparison with x,y>z. We can 
then approximate for V. It may be written 

-c 2 

v " "' 



where R is the distance OP, and 6 the angle between OP and 
the vector r = (x, j/, z) giving the position of the electron. This 
gives, for large J?, 

r cos 0\ 

+ ~R-}' 



V~- 



EXCITATION BY A PASSING PARTICLE 125 

Now r cos 6 is the projection of the vector (r, ?/, z) on OP; this 

in equal to , _. 

4 



Substituting, we find 

f 2 e*(vt.x+py) 

H*,!M;0 = - S -- 51 . 

where 7? - {(itf) 2 + ^ 2 }*. 

We must now use this form to calculate the transition proba- 
bilities from (6). Owing to the orthogonal properties (Chap. IV, 
8) of the functions 0,,, the first term in V gives no contribution. 
We see at once, integrating (6) from / = oo to / = -f oo, the 
whole time of the collision, that 



where x,, - U^x^dr, y,, = 

The 'matrix elements' x nQy etc. have already been introduced 
(Chap. IV, 10), and values given for the hydrogen atom. We 
have thus only to discuss the integration over t. 

If the impact parameter p is large compared with f/o> n0 , it 
will be seen that the integrand oscillates a large number of 
times in the range of t, of order p/v, in which it is not small. 
Under these conditions the integral is small; it may be shown 
to behave as exp( ^to n0 /v). Thus as a rough approximation 
we may assume that all collisions are ineffective for which p is 
greater than this value v/o n0 . 

This gives an interesting formula for the distance at which 
collisions cease to be effective in ionising or exciting an atom. 
If A ( = 2nc/aj nQ ) is the wavelength of the radiation (light) 
necessary to excite or ionise the atom, the critical distance is 

A v 

p^ , 

27TC 



so long as t'/c^l, so that no relativistic correction need be 
applied. 



126 TRANSITION PROB A B ILI Tl ES 

If, on the other hand, p is small compared with v/w nQ , the 
term exp(ia) l)0 t) can be neglected; the integral (8) then re- 
duces tot . 2 , . 2 
i 2 * 

' ~' 



Thus the chance P that the atom is excited into the state n by 
the passing particle is given by 

p 4* 4 l!/ B ol* , v 

" - + 9 9 ,- P<- - - () 



If N particles cross unit area per unit time, the chance per 
unit time that the atom is excited is 



H 2 v 2 j p 2 ' 

The upper limit of this integral may be taken to be ^/o> /<0 , and 
the lower a quantity of the order of the radius a of the atom, 
below which the approximations do not apply; thus for the 
mean chance per unit time that the atom is excited we obtain 



aoj 



j 

The loss of energy per unit length of path, dW/dx, for a par- 
ticle going through a gas containing N atoms per cm. 3 is thus 

dW 87rA T e 4 / v \ 

dx - -^^^ol^ol 1 ^ ), (10) 

where the summation is over all states into which a transition 
can occur. 

These formulae express the rate of loss of energy in terms of 
the quantity // /J0 , the 'dipole moment' of the transition to n. 

It is interesting to make an estimate of the rate of loss of 
energy without using quantum mechanics at all. At the 
moment /> the particle P exerts a force on an electron at O 
(Fig. 28) of which the component in the direction CO is 



f Put vtjp = tan 8. 



EXCITATION BY A PASSING PARTICLE 127 

The momentum, transferred to a free electron at O during the 
collision would thus be 



5 (^) + ;P* pv 
so that the electron if initially at rest would acquire energy 



The corresponding quantum mechanical formula for a bound 
electron is, from (9). 



where the summation is over all states n to which the electron 
can be excited without violating the condition aj nQ <v/p. Since 
we should expect that for near collisions the absorption of 
energy would be about the same as for a free electron, this 
suggests that the sum 



should add up to unity. This is in fact the case; the quantity 
/ n0 defined by 



f _ co o, |2 _ Tri^o , 

./ttO I #7<0 I -~ ~ I Jfii 



I2 



iiO I 



is known as the 'oscillator strength' of the transition to n, 
and, as already stated in Chapter IV, 10, 

S/,,o = I- 

Certain interesting results follow from (10). As the velocity 
v of an ionising particle passing through matter decreases, it 
will be seen that the rate of loss of energy increases, until 
v/w n0 becomes of the same order as a. It will be seen that 
this is the case when the velocity of the particle becomes 
comparable with that of the electron in the atom. When v is 
very small, although the atom may be very much perturbed 
by the passing particle during the collision, when the particle 



128 TRANSITION PROBABILITIES 

has gone away the wave function returns to its original form. 
The collision is then what we call adiabatic.f 

8. TRANSITIONS TO UNQUANTISED STATES 

A perturbing field, for example, the field of a passing charged 
particle or light wave, may ionise as well as excite an atom. 
Thus we have to consider transitions also to unquantised states. 
In order to use the formulae of the last section, the simplest 
procedure is to quantise the unquantised states by imagining 
the atom as enclosed in a large box of side L, say. For instance, 
suppose that the final state of the electron is represented by the 
wave function of a free particle J 

<Mr) = e'<*>. (11) 

If we introduce our box of side //, together with the boundary 
condition that i/r k shall be periodic with period L, then k has 

the values - _ . x , r 

k = ^(n^n^n^jL, 

where w 1 ,?z 2 , H 3 are integers. 

Therefore the number of states for which k lies between the 
limits Jt a and k^ + dk^k} and A' 2 -hd 2 , k 3 and Ar 3 -fdAr 3 is 

L^dk l dk^dk^[(27T)^. (12) 

Since the normalised wave function for the free electron is 



it follows that the chance that after time t the electron will be 
found in one of the states within the limits described by (12) is 

*dt. (13) 



o 



1 fe 



t For further references on the treatment of collisions by the method of 
impact parameters, see the following: Niels Bohr, 'The Penetration of Atomic 
Particles through Matter', K. Danske Vidensk. Selsk. xvm, no. 8, 1, 1948: 
Mott and Massey; E. J. Williams, Itcv. Mod. Phys. xvn, 217, 1945. 

J The IKS* of the free electron wave function (11) represents, of Bourse, an 
approximation. One ought to replace it by the wave function of an electron 
with positive energy moving in the iield of the atom. For details, cf. Mott 
and Massey, chap, xiv, $ 2-1. 



TRANSITIONS DUE TO A PERIODIC FORCE 129 

Alternatively one may require the probability that the electron 
is ejected in a direction lying in a solid angle <2Q and with 
energy lying between W and W + dW. It will easily be verified, 
since k = 27r<J(2mW)lh, that the number p(W)dW of states in 
this range is given by 



p (W)dW = 

= L*m z vdQdW/h*, (14) 

where v(= ^(2W/m)) is written for the velocity of the ejected 
electron. 

Thus the chance per unit time that the electron is ejected 
with energy between W and W + dW, and its direction of 
motion in the solid angle dfl y is 

m*vd&dW . , ... IT , , , 

- dt. 



, 1 f 

t - e 

\?tj 



4. TRANSITIONS DUE TO A FORCE ON AN ELECTRON 
WHICH IS PERIODIC IN THE TIME. 

This case is of great importance because it includes the action 
of a light wave on an atom. If v is the frequency with which 
the force changes, we shall show that transitions only occur 
with finite probability when the energy of the atom changes 
by hv. We have thus in this section to deduce from wave 
mechanics the well-known expressions of Bohr and Einstein for 
the absorption of energy by radiation, on which the quantum 
theory was first built up. 

It is convenient to write the perturbing term V(q; t) in the 
wave equation (3) in the form 

V = V Q e~ i<t}t + complex conjugate 

where V Q (q) is some function of the coordinates q of the electrons 
and o> = 2-rrv. Then (6) shows that the chance P n that, after 
time t, the atom is in the state n, is given by 

P I A tfM* 

r n 1 A n ( l ) I > 



130 TRANSITION PROBABILITIES 

where 



- = (n \ F 1 0)e"o-"<+ (n \ F * | 0) e*"">+<">< (15) 

i at 



and where (n \ V G \ 0) = \ifj* V Q ^ dr. 

This formula shows clearly that, unless to = o> n0 , A n does 
not increase with the time. The sudden application of the field 
at time t = will give a certain small probability for transitions 
into any excited state ;f but thereafter the probability will not 
increase unless the condition stated above is satisfied. But this 
condition may be written 

h v =\W n -W \. (16) 

We have shown therefore that transitions will only take place 
effectively if the Bohr frequency condition is satisfied. 

Suppose that W n > W$\ that is to say that we are dealing with 
a transition from a lower to a higher state. Then o> 7 , is positive; 
we have only to consider the first term in the right-hand side 
of (15); thus integrating with respect to the time, and putting 
in the condition that A n should vanish when t 0, we have 



Thus, squaring, 

p n (t) = p i ( i v 1 o) i- iriz^o- . (17) 

n ( W TiQ~~ w ) 

If we now set o> equal to o>,, , the probability P n (t) increases 
with the square of the time, and not linearly with the time as 
we should expect on physical grounds. This apparent contra- 
diction, however, can be resolved as follows. 

If the final state n is quantised, the concept of radiation of 
exactly the frequency given by (16) acting on the atom is in- 
correct. The absorption lines have a certain width; the problem 
of physical importance is to calculate the chance of excitation 

t The fact that the exciting radiation begins at a certain time means that 
its Fourier analysis must contain frequencies other than v. 



TRANSITIONS DUE TO A PERIODIC FORCE 131 

of the atom by radiation covering a band of frequencies 
wide compared with this. This is done in the next section, and 
it is shown that the probability increases linearly with the 
time. If the final state is unquantised (i.e. if the calculation 
refers to the ionisation of an atom by a light wave) we may 
use the fictitious quantisation of the last section. The atom is 
enclosed in a 'box' of side L, and the matrix element becomes 

(*IKJO) = ] 

The number of states p(W)dW with energies between W and 
W + dW is given by (14). The chance that after a time t an 
electron is ejected from the atom into the solid angle rfQ with 
any energy whatever is then given by the integral 

'p n (t) P (W n )dW nt 

the integral being over all energies W n . The quantity P n (t) 
defined by (17) has, however, a strong maximum when w n0 = aj, 
<ot being large; and thus for large t the integral becomes 



We have here put dW n Ma} n0 and taken outside the integral sign 

all terms except the one which gives rise to the sharp maximum. 

To evaluate the integral we put (ct> 7?0 o>) = x; we obtain 



f 00 2(1 -cosz) 
* I 2" 

J-oo X 



The integral is equal to 2?r; thus finally the chance that an 
electron is ejected with velocity v ( = Hk/m) lying in the solid 
angle rffi is 






132 TRANSITION PROBABILITIES 

3. EMISSION AND ABSORPTION OF RADIATION 

Planck in his theory of black-body radiation was the first 
to introduce the hypothesis that the energy of a light wave of 
frequency v is quantised, being a multiple of hv. Einstein 
explained the photoelectric effect by applying this idea to the 
interaction between radiation and matter; light can only give 
up its energy to matter in quanta of amount hv. Niels Bohr in 
his theory of the hydrogen spectrum introduced the comple- 
mentary idea, that an atom in making a transition from one 
quantised state to another would radiate quanta of energy of 
frequency determined by (16). 

Einstein was the first to give a correct quantitative descrip- 
tion of the intensities of radiative processes. Corresponding to 
the transitions between any two states n and ra of an atomic 
system, he introduced three probability coefficients, A nmt B n 



nm> 



and B mn . The coefficient A nm is defined as follows. A nm dt is 
the probability per time dt that an atom initially in the state n 
will make a transition to the state w, in the absence of all 
perturbation from outside. The coefficient A nm is supposed to 
be independent of t, of the past history of the atom and of the 
process by which it has been brought to the state n. 

The coefficients B mn and B mn are defined as follows. Suppose 
the atom is in the presence of radiation, unpolarised and inci- 
dent in all directions with equal intensity, and such that the 
energy per unit volume with frequency between v and v-f dv is 
I(v)dv. Let v nm be the frequency corresponding to a transition 
between the states n, m. Then if the atom is in the state m (the 
lower state) the chance per time dt that it will make a transition 
to the upper state n with the absorption of a quantum of 
radiation is B m>n l(v nm )dt. Also, if it is in the upper state n, the 
chance that in the presence of radiation it will make a transi- 
tion to the ground state with the emission of a quantum is 

{A nm + B nm I( Vnm )}dt. 

In attempting to calculate the A and B coefficients by means 
of wave mechanics we are faced with the following difficulty. 



ABSORPTION OF RADIATION 133 

The B coefficients may be calculated by a direct application 
of the analysis already given, the light wave being treated as 
an alternating electromagnetic field; so also may the proba- 
bility of the photoelectric effect, the ejection of an electron 
from an atom. The A coefficient, however, cannot be calcu- 
lated in this way; at first sight there seems to be no perturba- 
tion which will cause a spontaneous transition. 

A more elaborate theory than will be developed in this book 
is necessary to account for the A coefficient. Such a theory 
was first given by Dirac.f In this theory the radiation field 
is treated as a quantised vibrating system which interacts with 
the atoms, and, even when no radiation quanta are present, 
the interaction enables an excited atom to jump to a lower 
state and create one. For further details of this theory, the 
reader is referred either to the original papers or to more 
advanced text-books. J 

Fortunately, the A and B coefficients are connected by equa- 
tions which depend on thermodynamics. These are 

(19) 



nm mn ^ 

They may be proved as follows. Suppose that a number of 
atoms of the type concerned are in thermal equilibrium in an 
enclosed space at temperature 5P, together with the black-body 
radiation characteristic of that temperature. Then in a given 
interval of time as many atoms must make the transition 
upwards as downwards. Thus if I(v) refers to the intensity of 
black-body radiation, and N 1t ,N m are the numbers of atoms in 
the states n y m, 



m I( Vnm )} = N, H R IHn I( Vnm ). 
But by Boltzmann's law 



t P. A. M. Dime, Proc. Roy. A'oc. A, cxiv, 248, 1927. 

{ E.g., W. Heitler, quantum Theory of Radiation, 2nd ed., 1944. 



134 TRANSITION PROBABILITIES 

Now if we make the temperature tend to infinity, N n and N m 
become equal and /(v) becomes large; it follows that the two B 
coefficients are equal. We may therefore write, dropping the 
suffixes n, m, A \ 



We know, however, from the application of statistical mechan- 
ics to the radiation itself that the density of black-body radia- 
tion is quite independent of the type of atom present, and is in 
fact given by 



Comparing (20) and (21), (19) follows. 

We shall therefore, in the next section, limit ourselves to 
the calculation of B, the coefficient of absorption or forced 
emission. 

6. CALCULATION OF THE B COEFFICIENT 

We make use of the method of 4. The simplest approach 
will be to treat the light as an oscillating electric field E cos a)t 
and neglect the effect of the magnetic field on the electron. 
A more rigorous approach which gives approximately the same 
result will be mentioned at the end of this section. 

We take the electric vector E along the z-axis; then the 
potential energy of the electron is 

V(x, y, z; t) = eEz = eE Q z cos cut. (22) 

This, as in 4, can be written 

F = F e-^ + F *e^, 
where V Q = F * = &E Q z. 

Thus (15) can be integrated, subject to the condition that 
Ai(0 = O when t = 0, to give 



, 

' t 

*w MO -fco 

(23) 



THE B COEFFICIENT 135 

where z n0 denotes the matrix element (already introduced on 
pp. 75, 125) 



We now consider the two terms within the square brackets. 
If CD is not equal or very nearly equal to o> n0 , the function 
oscillates with t\ it does not increase. If aj o> n0 vanishes, how- 
ever, it increases with t. Thus, if we are dealing with absorption, 



so that the initial state is below the final and o> 7{0 ( = W n 1 
is positive, we need consider only the former term and values 
of a) near to o> w0 . In the case of stimulated emission the initial 
state of energy W is above the final state, co nQ is negative and 
only the second term need be considered. 

In the former case, that of absorption, we retain the first 
term in (23). Taking the square, we find 

I 1 /A 12 f ^^n^Hft i **\ * COS \ 
I A n (t) \ = 



This function, giving the chance that after time t the atom is 
in the state n, has as we expect a strong maximum for w n0 to. 
To obtain an expression for the absorption coefficient, we sup- 
pose that the atom is irradiated with light such that the energy 
density between the frequencies y, v + dv is I(v)dv. The energy 
density in the light wave with electric vector E Q cosa>t is the mean 
value of (E* + # 2 )/87r, or of # 2 /47i and thus #2/877. Replacing E* 
by $irl(v)dv in the expression above for | A n (t) | 2 , we see that the 
chance that after time t the atom is excited into the state n is 



p = 2, 

* 



\ 2 r / 

/ J 



Owing to the strong maximum at o> o MO , we may take I(v) 
outside the integral sign. The substitution of p. 131 enables the 
integral to be evaluated. Writing 

dv = do>/27r, (o> w0 o>) t = x, 
we obtain P = 



136 TRANSITION PROBABILITIES 

The integral as before is equal to 2ir\ thus 
P 



(py 
^ 



which increases with t as it should. 

To obtain the B coefficient we must average over all direc- 
tions of polarisation; we obtain 

B = IT P ( ' x |2 + ' y P + ' * |2) ' (24) 

The A coefficient may be obtained from (19), and is 

Ko! 2 + l3/,J 2 + |z n ol 2 ). (25) 

It is of interest to express this in terms of the oscillator 
strength already introduced on pp. 76, 127; we find, since 
normally all but one of the three matrix elements in (25) will 
vanish, 



If one inserts numerical values one finds 

A = 2-2 x 1C 8 -^ sec." 1 , 

where A is the wavelength in ft. For a line of strong intensity 
/ lies, say, between 0-1 and 1. Thus the transition probability 
is about 10 8 sec.~ 1 for a line in the visible region, but much more 
for lines in the X-ray region. 

Finally we must add a few remarks about the approximation 
(22) for the field of the light wave. Here we neglect both the 
effect of the magnetic field, and the variation of E within the 
atom. The correct perturbing term is actually, for a light wave 
moving along the z-axis with electric vector along the z-axis 

complex conjugate 
r 



CHAPTER VII 

RELATIVISTIC AND NUCLEAR 
DEVELOPMENTS 

1. DIRAC'S RELATIVISTIC WAVE EQUATION 

No fill] account of Dirac's relativistic theory will be attempted 
here. In his original paperf on the subject Dirac showed that, 
in order to write down a wave equation linear in the time (it- 
must be linear for the reasons given in Chap. Ill, 2), and 
satisfying the principle of relativity, it was necessary to ascribe 
to the electron a fourth degree of freedom, or spin; he showed 
that, if this were done, the properties previously ascribed to 
the spin could be deduced without further assumptions from 
the equation. In any relativistic treatment x,y,z must be 
treated on the same footing as it; thus the equation is linear in 
djdx, 8/dy, djdz as well as d/dt. The state of the electron is des- 
cribed in Dirac's theory by four wave functions <f/ ly </r 2 > 0a> 04 > an d 
the quantity j i/r | 2 of the non-relativistic theory is replaced by 

0101 +020? +030? + 0404- 

The relativistic theory should be used in all cases where the 
velocity of the electron approaches that of light, and also, even 
when this is not so, to calculate the separation of spin doublets. 
Some of the more important applications are the following: 

(a) Calculation of separation of the ns and np states of 
hydrogen (they coincide only in the non-relativistic approxi- 
mation) and the spin doublet separation of the np state. f 

(b) Calculation of the separation between L 1 and L u levels 
in the X-ray spectrum. These levels have quantum numbers 

t P. A. M. Dime, Proc. Roy. Soc. A, cxvn, 010, 1028. Cf. also Quantum 
Mechanics. 

I Dirac, Proc. Roy. Soc. A, cxvn, 010, 1928, and Quantum Mechanics, 
1947, p. 208; H. liethc, Handb. Ptiys. xxiv, pt. 1, 311, 

Cf. H. Bethe, Handb. Phys. xxiv, pt. 1, 822, 1988. 



144 RELATIVISTIC DEVELOPMENTS 

2s, 2p. The separation is here partly due to the relativistic 
correction, comparatively large for heavy nuclei since the mean 
velocity of the electron increases with Z, and partly to the fact 
that the field no longer has the Coulomb form owing to the 
screening effect of the other electrons. 

(c) Calculations of the photoelectric effect and Cornpton 
effect for high energies of the electrons. f 

(d) Calculations of the scattering of fast electrons by atomic 
nuclei, and of the polarisation of the electron beam produced 
by such scattering. In the scattered beam it may be shown 
that the spins are no longer oriented at random 4 

(e) Scattering of fast electrons by electrons. 

2. THE POSITRON OR POSITIVE ELECTRON 

The positive electron was discovered independently by 
Anderson|| and by Blackett and Occhialini.^f It is a particle 
with the same mass as the electron, but carrying a positive 
charge. It is unstable, being produced when fast electrons or 
y-rays interact with nuclei, and being capable of combining 
with an electron to produce one or more y-ray quanta. 

The theory given by Dirac to account for the positron is as 
follows. According to relativistic theory, the relation between 
the energy W of a particle of mass m and its momentum p is 

W 2 = c 2 (m 2 c 2 -fp 2 ). (1) 

If p is small this gives 

W = we 2 -f- $p 2 lm -f terms in (1/c 2 ), 

so that the formula for the energy includes the energetic equiva- 
lent me 2 of the mass as well as the kinetic energy p 2 l2m. 
Formula (1) gives for W 

W = c^(m 2 c 2 +p 2 ). 

f A discussion of this subject is given by W. Heitler, Quantum Theory of 
J Mott and Mussey, chap. iv. [Itadiation. 

Mott tmd Mussey, p. 365. Original paper, C. Moller, Z. Phys. LXX, 786, 1981. 
J! C. D. Anderson*, Phys. Rev. xu, 405, 1932. 

f P. M. S. Rlackett and G. P. S. Occhialini, Proc. Roy. Soc. A, cxxxix, 
99, 1933. 



THE POSITRON 145 

In relativistic theory before the advent of wave mechanics it 
was possible to limit the physically significant values of W to 
the positive square root only; W could only change continuously 
and would not be able to pass through the forbidden values 
between me? to reach the negative values. In wave mechan- 
ics, on the other hand, the theory allows a particle to jump 
from any one allowed state to any other under the influence 
of a suitable perturbation. A suitable perturbation is the com- 
bined influence of radiation and of an atomic nucleus. Nothing 
corresponding to such transitions appeared to occur in nature; 
in fact it was not clear before the discovery of the positive 
electron exactly what significance the states of negative energy 
might have. 

To overcome this difficulty in the theory, Diracf made the 
following proposal. He suggested that the states with negative 
energy should all be occupied, in the same sense that the states 
of the Fermi distribution of a metal are all occupied (Chap. V, 
7-3). Space is thus to be filled with a uniform 'gas' of electrons 
of infinite density, and with a uniform distribution of positive 
charge to neutralise their charge. Or at any rate space was 
assumed to behave as if this were so. 

The immediate consequence of this hypothesis is that a 
quantum of radiation, or a charged particle of kinetic energy 
greater than 2mc 2 , is capable in suitable circumstances (for 
example, in collision with a nucleus), of lifting an electron 
from the continuum into a state of positive energy; one would 
thus observe the creation of 

(a) an electron, 

(6) a vacancy in the continuum, which OppenheimerJ first 
showed would behave like a positive electron. 

As is well known, the creation of such pairs is frequently 
observed. They can be created for instance by X- or y-rays of 
frequency v such that 

l\v> 2mc 2 , 

t Cf. Quantum Mechanics, p. 272. 

t J. R. Oppenheimer, Phys. Kev. xxxv, 939, 1930. 

10 



146 RELATIVISTIC DEVELOPMENTS 

as will be clear from Fig. 29. They cannot be formed in free 
space by a single quantum, since it is impossible to conserve 

at the same time energy and 
___ momentum; they can, however, 

I ? be formed when radiation falls 

2 _ lnc _ ______ zero on matter. The calculation of 

the intensity of pair formation 
by X-rays is carried out by the 
method of Chapter VI, 8; the 
^ Fig. 29. initial and final wave functions 

are those for an electron, with 

positive and negative energy values respectively, moving in 
the Coulomb field of the nucleus. 



3. APPLICATIONS TO NUCLEAR PHYSICS 

Applications of wave mechanics to the nucleus can only be 
made in certain special cases, because of our ignorance of the 
nature of the forces between the nucleons (protons and neu- 
trons) of which the nucleus is believed to be made up. Problems 
to which wave mechanics can be applied are of three main types: 

(a) Problems where, for one reason or another, only the 
Coulomb part of the field round a nucleus is important. In this 
category we may place the theory of a-decay, of the internal 
conversion of y-rays and perhaps the theory of /3-decay. 

(6) Problems in which some assumption is made about the 
force between nucleons, and wave mechanics is used to calculate 
the consequences of the assumption. Attempts have been made 
along these lines to calculate the binding energy of the deuteron 
(one proton with one neutron), and of heavier nuclei. 

(c) Problems, such as those concerned with the compound 
nucleus and the spacing between the energy levels of the 
nucleus, which do not depend on the law of force assumed. 

In this book we shall discuss briefly only two of these prob- 
lems, the theories of a- and of /?-decay. The first is the best 

J For details of the calculation, see W. Heitler, chap. iv. 



THEORY OF ALPHA-DECAY 147 

known illustration of the quantum mechanical tunnel effect, 
the second gives an illustration of how, in a field still not 
completely understood, it is possible to obtain useful results 
from wave mechanics. 

3-1. The theory of oc-decay of radioactive elements 

The quantitative explanation by Gamowf and by Gurney 
and Condon J of the emission of a -particles from radioactive 
nuclei was one of the earliest successes of wave mechanics. 
The facts to be explained are as follows. Given N atoms of a 
radioactive substance, NXdt of them will disintegrate spon- 
taneously in a time interval dt, where A, the decay constant, is 
independent of the time, and thus independent of the age of 
the nucleus. For different radioactive substances A has a very 
wide range of values from c. 10~ 18 sec." 1 to c. 10 6 sec." 1 ; there 
exists a roughly linear relationship between log A and the energy 
W with which the particle is emitted. 

A nucleus of mass number M + 4 (i.e. containing M + 4 nu- 
cleons) and of atomic number Z+2 will show a-decay if and 
only if energy is released on removing an ex-particle to form a 
nucleus (M, Z): we shall assume this to be the case, the energy 
difference being W. Consider then the potential energy V(r) of 
this a-particle at a distance r from the product nucleus ( M, Z). 
At large distances 



at small distances, comparable with the radius of the nucleus, 
V(r) must take a form corresponding to an attractive force, 
because the a-particle is held within the nucleus for a long time. 
The form of the potential energy may be as in Fig. 30, curve (b): 
we have, of course, no detailed knowledge of the attractive part 
of the curve, and indeed the a-particle may be supposed to lose 
its identity within the nucleus. 

t G. Gamow, Z. P%.v. j,xi, 204, 1928. 

t R. W. Gurney and K. U. Condon, Mature, Lond. cxxu, 4JJ9, 1928; Pfiys. 
Rev. xxxin, 127, 1929. 

10* 



148 



RELATIVISTIO DEVELOPMENTS 



The a-particle eventually escapes with positive energy W\ 
within the nucleus it may be pictured as having the same 
energy W. Thus the a-particle within the nucleus is separated 
from the outside by the 'potential barrier* AOB of Fig. 30; it is 
possible for the a-particle to penetrate this barrier by tunnel 




Fig. 30. Potential energy of an a-particle in the field of a nucleus. 

effect. Thus the decay constant A, the chance per unit time 
that the a-particle escapes, is given by 



where v is the velocity of the a-particle within the nucleus, rf is a 
quantity of the dimensions of its diameter and P the chance that 
a particle incident on the potential barrier should pass through 
it. The quantity v/d gives an estimate of the number of 
times per second that the a-particle impinges on the potential 
barrier; no very accurate estimate need be made of it, since 
P varies very rapidly indeed with the various parameters. 



THEORY OF ALPHA- DECAY 149 

We set v~ 10 9 cm. /sec., rf ~ 10~ 12 cm., and obtain 

A= 10 21 Psec. - 1 . 

P may be calculated by the W.K.B. method (Chap. II, 6); 
equation (10) of Chapter II gives 

(2) 

m is here the mass of au oc-particle, and the integration is from 
A to B in Fig. 30. 

A convenient approximation for evaluation of this integral 
is to replace V(r) by the form shown in Fig. 30, curve (c), 
namely 

V(r) = ^~ r>r , 

= const. r < r , 

r being a 'nuclear radius'. It is not suggested that V(r) will 
actually have this form, but our complete ignorance of the true 
form of V(r) within the nucleus, or of whether the interaction 
can be represented by a potential energy function at all, makes 
it as good an approximation as any other. Then (2) becomes 



where r x = 2Ze*/W. 

The integral may be evaluated by setting 
cos 2 14 = r/r,; 

/ 9-1 

we find InP = - 

where cos 2 w = ^"o/ r i- Since rj/^ is small, we may write 

and therefore 



150 RKLATIV1STIC DEVELOPMENTS 

It will be seen that r l and Z are the most important factors in 
determining P, and hence the decay constant; these, obviously, 
define the size of the potential barrier through which the tunnel 
effect occurs. 

Substituting for r l9 we see that a linear relation exists be- 
tween InP (and hence In A) and ^W or v, the velocity with 
which the a-particle emerges. The relation may be written 

log 10 A = ^- --- + CVr , 
where A = 21, 



: 

B = _- -~ 1-2 x lO'cm./sec.- 1 . 
Tun 1 



.- 

Bin 10 

The relationship between log A and v, the velocity of the emitted 
a-particle, explains the empirical Geiger-Nuttall law, according 
to which the decay constants of radioactive elements increase 
exponentially with the energy. Also the very large variation 
in decay constants, from 10 5 sec.~ 1 (thorium C) to 0-5 x 10~ 18 
sec. ~~ 1 (uranium) are explained by the large variation of the 
term BZ/v for velocities varying between about 1-5 and 
2 x 10 9 cm. /sec. 

Detailed comparison of observed decay constants and veloci- 
ties with the formula enable estimates of r , the nuclear radius, 
to be obtained. The values deduced lie between 0-5 and 
1-0 x 10- 12 cm.f 

3-2. The theory of fi-decay 

In Gamow's theory of a-decay explained in the last section, 
the decay constant A is in principle deduced from constants 
already known, namely, the atomic number Z, the energy of 
the emitted particle, its mass m, e and H. The unknown nuclear 

f For further details of the theory of radioactive decay, see G. Gamow, 
Atomic Nuclei and Nuclear Transformations, 3rd ed., p. 174, Oxford, 1949. 



THEORY OF BETA-DECAY 151 

radius enters only as a small correction. To develop a theory 
of /?-decay, on the other hand, one has to make a new assump- 
tion; this is that a neutron will change into a proton with the 
creation of an electron and a neutrino if energy is gained 
thereby; or that a proton will change into a positive electron 
and a neutrino, again if energy is gained. And also one assumes 
that the probability that such a transition occurs, the electron 
and neutrino appearing in given quantised states with wave 
functions ifj e (x,y,z),i)j n (x,y,z), is proportional to | i/^i/r,, | 2 , both 
functions being taken at the point where the nucleon is. This 
theory, due originally to Fermi, f cannot account for the abso- 
lute magnitude of the decay constant; a new constant of nature 
has to be introduced for this; but it can account for the varia- 
tion of the decay constant from element to element, and for 
the shape of the /?-ray spectrum. 

One speaks of the creation of an electron for the following 
reason. It is impossible to envisage an electron as within the 
nucleus before it appears outside; the radius of the nucleus is 
of the order 10~ 12 cm., so the wavelength of the electron would 
have to be of this order; it will easily be seen that such a 
wave-length corresponds to c. 10 8 eV, which is much greater 
than the energies of c. 10 6 eV with which they are actually 
emitted. 

The neutrino is a particle with no mass or charge moving 
with the velocity of light. As for a light quantum, its kinetic 
energy W tl and momentum p fl are related by the relativistic 
equation, which follows from (1) when m is put equal to zero, 

W n = cp a . 

It is supposed to have a spin \H, like an electron. It is intro- 
duced into the theory primarily in order to account for the 
continuous )8-ray spectrum; it is supposed that a beta-active 
nucleus has a given amount of energy W Q to dispose of when 
the neutron changes into a proton, but that this may be 
distributed between the electron and neutrino in any way. 

t E. Fermi, Z. Phyx. i.xxxvm, 101, 11KJ4. 



152 U EL ATI VIS TIC DEVELOPMENTS 

For small atomic numbers Z and high enough energies, one 
can describe the electron after emission by a plane wave. The 
neutrino can be so described in any case. Thus if the whole 
system is supposed shut up in a box of volume V as in Chapter 
VI, 3, both *ft t and $ n are of the form 



Taking the nucleus at the origin, we see that if* e , I/J H at that point 
are both independent of p. Thus the chance per unit time for 
the creation of an electron and neutrino is the same for all 
states in which they may be found. 

Now the number of states of the electron with energies 
between W c and W f + dW e is proportional to 



where p e is the momentum of the electron. With each of these 
states is associated a number of states of the neutrino pro- 
portional to p\. Thus the chance that the electron is emitted 
with energy between W e and H + dH is proportional to 



(3) 

Now for p"^ we write p% 



where W Q as before is the energy available for the reaction. 
p f is given in terms of W e by the relativists c formula 



so that (3) becomes 

const. e(e 2 - 1 )* ( - c) 2 de, (4) 

where e = WJmc 2 and e = W0/mc 2 . 

This formulaf gives the dependence of the number of emitted 
/J-particles in the continuous spectrum on the energy . 

t For a comparison of (4) with experiment, see, for example, H. Bethe, 
Elementary Nuclear Theory: A Short Course on Selected Topics; or G. Gamow 
and C. L. Oitchfield, Atomic Nuclei and Nuclear Transformations^ 3rd ed., 
Oxford, 1949. 



BIBLIOGRAPHY 

1. General treatises on quantum mechanics, dealing with the 
applications rather than with foundations: 

N. F. Mott and I, N. Snccidon, Wave Mechanics and /As- Applications, 
Oxford. 1948. 

L. I. Schiff, Quantum Mechanics, New York, 1949. 

Certain chapters in J. C. Slater and N. H. Frank, Introduction to 
Theoretical Physics, New York, 1933. 

Certain chapters in F. K. Richtmeyer and E. II . Kennard, Introduction 
to Modern Physics, New York, 1947. 

The article by II. Betlie, 4 The Quantum Mechanics of the Problems of 
One and Two Electrons' ('Quantenmechanik der Ein- und Xwei-Elektron 
probleme') in Handbuch der Physik, vol. 24, pt. 1, 1938. This ghcs a 
very complete account of the theory of the hydrogen and helium atoms, 
the hydrogen molecule and their interaction with radiation. 

2. General treatises on quantum mechanics, dealing with 
fundamentals: 

P. A. M. Dirac, Quantum Mechanics, 3rd cd., Oxford, 1947. 

The article by VV. Pauli, 'General Principles of Wave Mechanics 1 ( 4 Alge- 
meine Prinzipien der Wellenmechanik') in Handbuch der Physik, vol. 24, 
pt. 1, 1933. 

3. Books dealing with particular applications: 

ATOMIC SPECTRA 

E. V. Condon and G. II. Shortley, The Theory of Atomic Spectra, 
Cambridge, 1931. 

INTERACTION UET\\EKN MATTER AND RADIATION 

W. Heitler, Quantum Theory of Radiation, 2nd ed.. Oxford, 1944. 

COLLISIONS BETWEEN ELECTRONS AND ATOMS 

N. F. Mott and H. S. W. Massey, Theory of Atomic Collisions, 2nd cd., 
Oxford, 1949. 

SOLID STATE AND MAGNETISM 

F. Seitz, Theory of Solids, New York, 1942. 

N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, 
Oxford, 1940. 

N. F. Mott and H. Jones, Theory of the Properties of Metals and Alloys, 
Oxford, 1930. 



154 BIBLIOGRAPHY 

SOLID STATE AND MAGNETISM [continued] 

\V. Hunie-Rothory, Atomic Theory for Students of Metallurgy, Institute 
of Metals, Monographs and Reports Series, 1946. 

K. C. Stoner, Magnetism and Matter, London, 1934. 

G. V. Raynor, An Introduction to the Electron Theory of Metals, Institute 
of Metals, Monographs and Reports Series, 1947, 

NUCLEAR PHYSICS 

G. Gamow and C. L. Critchlicld, Theory of Atomic Nucleus and Nuclear 
Energy Sources, Oxford, 1949, being the third edition of Gamow's Atomic 
Nuclei and Nuclear Transformations. 

H. Bethc, Elementary Nuclear Theory. A Short Course on Selected 
Topics,. New York, 1947. 

MOLECULAR Fouccs 

J. C. Slater, Introduction to Chemical Physics, New York, 1939. 
\V. G. Penney, The Quantum Theory of Valency, Methuen's Chemical 
Monographs, 1935. 

J. II. van Vleck and A. Sherman, 'The Quantum Theory of Valency \ 
article in Reviews of Modern Physics, vu, p. 167, 1935. 

G. B. 15. M. Sutherland, Infra-red and Raman Spectra, Methiien's 
Physical Monographs, 1935. 



INDEX 



Adiabatic collisions, 128 

Alkali-halides, 107 

Alpha-decay of nuclei, 147 

Alpha particles, scattering in helium, 

97 

Antisymmetrical wave functions, 92 
Atomic scattering factor, 105 
Auger effect, 130 

Band spectra, 54 
Be&sePs equation, 5 
Beta-decay of nuclei, 150 
Bnhr's magneton, 80 
Bohr's quantum theory, 58 
Born's interpretation of the wave 
function, 40 

Collision problems, 30, 123, 142 
Conduction bands in solids, 114, 116 
Conservation of charge, 27 
Covalent forces, 107 

Davisson and Germer, experiments 

of, 21 

De Broglie wave-length, 22 
Dielectric constant, 77 
Diffraction of electrons, 21 
Dirac's radiation theory, 133 
Dime's theory of spinning electron, 

143 
Dispersion formula, 76 

Effective mass of electrons, 115 
Einstein A and B coefficients, 132 
Exchange integral, 100, 113 

Fermi-Dirac statistics, 110 
Fermi's theory of /3-decay, 151 
Franck-Condon principle, 91 

Gamow's theory of a-decay, 147 
Geiger-Nuttal law, 150 
Gerlach and Stern's experiment, 81 
Group velocity, 12, 43 
Gyromagnetic effect, 82 



Hamiltonian operator, 46 
Hartree's method of self -consistent 

fields, 68, 106 

Heisenberg's uncertainty principle, 5( 
Helium atom, 68, 70, 00 
Hydrogen, atomic spectrum, 62 
Hydrogen atom, motion of proton in 

54, 87 

Hydrogen molecular ion, 108 
Hydrogen molecule, 112 
rotational states, 54, 02 

Lcgendrc polynomials, 10 
London-Heitler method, 112 

Mean free path of electrons in metals 

115 

Methane, wave functions for, 114 
Molecular orbitals, 112 
Molecular spectra, 88 
Morse potential, 02 

Neutrino, 151 
Nuclear radius, 150 

Orthogonal property of wave func 

tions, 70 

Orthohelium, 101 
Orthohydrogen, 08 
Oscillator strengths, 76, 127, 136 

Parahelium, 101 
Parahydrogen, 08 
Pauli principle, 06, 103 
Photoelectric effect, 138, 144 
Planck's radiation theory, 132 
Polarisibility of atoms, 74 
Positron, 144 

Reflection of waves, 8, 30 
Refraction of waves, 11, 27 
Refractive index, imaginary value; 

of, 10 
Relativity, principle of, 83, 143, 144 

151 



156 



INDEX 



Rotational states, 54, 88 
Rutherford's scattering formula, 38 

Scattering of electrons by atoms, 36, 

142 

Scattering of light by atoms, 77 
Selection rules, 137 
Self-consistent field, method of, 08, 

106 

Semi-conductors, 110 
Silicon, 117 

Simple harmonic oscillators, 58 
Spin doublets, 83 
Spinning electron, 81 

Dime's theory of, 143 
Stark effect, 78 
Strong field emission, 120 
Symmetrical wave functions, 9- 



Thomas-Fermi method, 106 
Thomson, G. P., experiments of, 21 
Transition probabilities, 136 
Tunnel effect, 32, 120, 148 



Van der Waals' forces, 106 
Variational methods, 69 



Wcntzel-Kramers-Brillouin method, 

3, 28, 35, 59, 66, 148 
Work function, 119 



X-ray spectra, 54, 103, 119 



Zeeman effect, 79